Differential topology with a view to applications [1 ed.] 9780273002833

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Research Notes in Mathematics

D R J Chillingworth

Differential topology with a view to applications

Pitman LONDON SAN FRANCISCO-MELBOURNE

9

Titles in this series 1 Improperly posed boundary value problems A Carasso and A P Stone 2 Lie algebras generated by finite dimensional ideals 1 N Stewart 3 Bifurcation problems in nonlinear elasticity R W Dickey 4 Partial differential equations in the complex domain D LColton 5 Quasilinear hyperbolic systems and waves A Jeffrey 6 Solution of boundary value problems by the method of integral operators D LColton 7 Taylor expansions and catastrophes T Poston and I N Stewart 8 Function theoretic methods in differential equations R P Gilbert and R J Weinacht 9 Differential topology with a view to applications D R J Chillingworth 10 Characteristic classes of foliations H VPittie 11 Stochastic integration and generalized martingales A U Kussmaul 12 Zeta-functions: An introduction to algebraic geometry A D Thomas 13 Explicit a priori inequalities with applications to boundary value problems V G Sigillito 14 Nonlinear diffusion W E Fitzgibbon III and H F Walker 15 Unsolved problems concerning lattice points J Hammer 16 Edge-colourings of graphs S Fiorini and R J Wilson 17 Nonlinear analysis and mechanics: Heriot-Watt Symposium Volume I R J Knops 18 Actions of finite abelian groups C Kosniowski 19 Closed graph theorems and webbed spaces M De Wilde

Differential topology with a view to applications

NUNC COGNOSCO EX PARTE

THOMAS J. BATA LIBRARY TRENT UNIVERSITY

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/differentialtopoOOOOchil

D R J Chillingworth University of Southampton

Differential topology with a view to applications

Pitman LONDON • SAN FRANCISCO • MELBOURNE

QUI3. &

. C v?

/1?6b

PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB FEARON-PITMAN PUBLISHERS INC. 6 Davis Drive, Belmont,California 94002,USA

Associated Companies

Copp Clark Ltd, Toronto Pitman Publishing New Zealand Ltd, Wellington Pitman Publishing Pty Ltd, Melbourne First published 1976 Reprinted 1977 Reprinted 1978 AMS Subject Classifications: (main) 58AOS, 34-02 (subsidiary) 26A57,34C-, 34D30, 58F-, 70-34 Library of Congress Cataloging in Publication Data Chillingworth, David. Differential topology with a view to applications. (Research notes in mathematics; 9) Includes bibliographical references. 1. Differential topology. I. Title. II. Series. QA613.6.C48 514'.7 76-28202 ISBN 0-273-00283-X ©DR J. Chillingworth 1976 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent,resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. Reproduced and printed by photolithography in Great Britain at Biddles of Guildford

Preface With the increasing use of the language and machinery of differential topology in practical applications, many research workers in the physical, biological and social sciences are eager to learn some of the background to the subject, but are frustrated by the lack of any self-contained treatment that neither is too pure in approach nor is written at too advanced a level. Few people have the time and energy to embark on a systematic study in a field other than their own, and so this book is written with the purpose of making differential topology accessible in one volume as a working tool for applied scientists.

It could also serve as a guide for graduate students

finding their way around the subject before plunging into a more thorough treatment of one aspect or another.

The book should perhaps be read more

in the spirit of a novel (with a rather diffuse ending) than as a text-book. The particular aim is to study the global qualitative behaviour of

dynamical systems, although there are numerous byways and diversions along the route.

A dynamical system is some system (economic, physical,

biological ...) which evolves with time.

Given a starting point,

the system

moves within a universe of possible states according to known or hypothesized laws, often describable locally by a formula for the 'infinitesimal* evolution, namely a differential equation.

The global

theory is the theory of all possible evolutions from all possible initial states,

together with the way these fit together and relate to each other.

Qualitative theory is concerned with the existence of constant (equilibrium) behaviour, periodic or recurrent behaviour, and long-term behaviour.

313734

together with questions of local and overall stability of the system.

Global

qualitative techniques, mainly stemming from the work of Henri Poincard^ (1854-1912), are important both because precise quantitative theoretical solutions may in general be unobtainable, and because in any case a qualitative model is the basis of a sound mental picture without which mechanical calculation is highly dangerous. The natural universe of evolution for a dynamical system is often a differentiable manifold; the evolution itself is a flow on the manifold, and a differential equation for infinitesimal evolution becomes a Vector field on the manifold.

Chapters 1-3 of the book are concerned with defining

and explaining these terms, while Chapter 4 goes into the qualitative theory of flows on manifolds, ending with some discussion of bifurcation theory. There is an Appendix on basic terminology and notation for set theory. Inevitably there are many topics which should have been included or developed but which would have expanded the volume to twice its size. Differential forms are hardly mentioned, singularity theory is only touched upon, and the fascinating terrain of general bifurcation theory for differential equations, including the Centre Manifold Theorem (one of the few really -practical applications of differential topology), is left largely unexplored.

I hope the tantalized reader will be able to follow up these

topics via the references given. Formal prerequisites are kept to a minimum.

The ideas from topology and

linear algebra that are needed are mostly developed from first principles, so that the basic requirements are hardly more than a familiarity with derivatives and partial derivatives in elementary calculus - although these, too, are defined in the text.

The exceptions to this are complex

numbers, which are assumed to be well-known objects to mathamatically-minded

scientists, and determinants and eigenvalues of matrices which may be less well-known to some but are everyday equipment for others.

My excuse for

this logical inconsistency is lack of space and the need to draw a line somewhere:

I felt that it was more important to discuss carefully some

of the fundamental ideas about linear spaces upon which the rest of the structure is built than to go on to techniques familiar to many people and in any case quite accessible elsewhere.

In the first three Chapters the

complex numbers feature mainly in examples and illustrations but in their roles as eigenvalues they become crucial to the main plot in Chapter 4. As overall references for the qualitative theory of dynamical systems, I suggest the now historic survey article by Smale very readable lecture notes of Markus differential equations by Amol'd

[73 ].

[125] and the subsequent

The excellent books on

[ll] and Hirsch and Smale

[55]

directed towards the qualitative theory of flows on manifolds.

are both

For back¬

ground on differential topology a recent and attractive text is Guillemin and Pollack

[48J:

there is also a forthcoming book by Hirsch [52 ].

The

fascinating article on applications to fluid mechanics and relativity by Marsden, Ebin, and Fisher

[]75j

is highly recommended (see also the

introduction to differential topology by Stamm in the same volume). The present book grew from a series of lectures given to a mixed audience of pure and applied mathematicians, engineers, physicists and economists at Southampton University in 1973/74.

It is through the

encouragement of several of these colleagues that I have expanded the lecture notes into book form, and I am grateful to them and others for helpful comments and criticisms.

I am particularly indebted to Peter Stefan

of the University College of North Wales, Bangor who carefully read the original notes and offered many detailed suggestions for improvement.

* Now appeared: highly recommended

Despite all this assistance, I claim the credit for errors.

I would also

like to thank Professor Umberto Mosco and Professor Nicolaas Kuiper for hospitality at the Istituto Matematico dell' University di Roma and the I.H.E.S., Bures-sur-Yvette, respectively, during visits to which I wrote up much of the notes.

I am grateful also to Pitman Publishing for their

interest in the book and patience during its production,

to my wife Ann for

tolerating the side-effects, and especially to Cheryl Saint and Jenny Medley for spending many long hours producing such a perfect typescript. Finally, my special thanks go to Les Lander for taking upon himself the task of drawing all the figures in the book, and obtaining such professional results in a short space of time.

David Chillingworth Southampton, August 1976.

I am grateful to Mike Irwin, Mark Mostow and my father H.R. Chillingworth for kindly pointing out a number of misprints and errors which, fortunately, I have been able to correct for the second printing of the book. Also I would like to take this opportunity to draw attention to the Appendix, which explains some set-theoretic terminology and notation that is used without comment in the text.

D.R.J.C. May 1977.

Contents 1. Basic topological ideas 1.1 1.2 1.3 1.4 1.5 1.6 1.7

The concept of a function Continuity Continuity from a more general viewpoint Further topological concepts Homeomorphism of spaces and equivalence of maps Compactness Connectedness

Remarks on the literature

1 4

9 18 25 27 33

35

2. Calculus 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Differentiation Linear spaces and linear maps Normed linear spaces Differentiation (continued) Properties and uses of the derivative Higher orders of differentiation Germs and jets Local structure of differentiable maps I: Non-singular behaviour 2.9 Local structure of differentiable maps II: Singularities

36 38 45 53 64 72 80

100

Remarks on the literature

115

92

3. Differentiable manifolds and maps 3.1 The concept of a differentiable manifold 3.2 Remarks, comments and more examples of differentiable manifolds 3.3 The structure of differentiable maps between manifolds 3.4 Tangent bundles and tangent maps 3.5 Vector fields and differential equations

116 131 146 162 179

Remarks on the literature

189

4. Qualitative theory of dynamical systems 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Flows and diffeomorphisms Local behaviour near fixed points and periodic orbits Some global behaviour Generic properties of flows and diffeomorphisms Global stability Dynamical systems under constraint Breakdown of stability: bifurcation theory

Remarks on the literature

191 204 217 221 227 245 252 267

APPENDIX: Terminology and notation for sets and functions

268

REFERENCES

273

Index

284

1 Basic topological ideas

1.1

THE CONCEPT OF A FUNCTION

We usually visualize a real-valued function of a real variable in terms of its graph:

For a given number anothet real number coordinatized by

x

(i.e.

f(x).

(x,y)

x e R)

the function

f

provides

The graph consists of all points in a plane

which satisfy

graph(f) = ((x,y)

y = f(x),

or in formal notation

e R x R | y = f(x)}

.

Another 'picture' of the, function, though less useful than the above, is the following:

R

R

x' fCx) x

f(x') Figure 2

1

For a real function

f

of two real variables

be thought of as a landscape, with 'horizontal' plane and Formally, we have

f(x^,x2)

(x^,x2) e

(x^,x2)

graph(f) = {((x1,x2), y)

Alternatively, we can picture

f(x^,x2)

e R" x R = R3

f

'vertically'.

2 ,

the graph could

as coordinates in a

being measured

R x R = R

(x^,x2)

e

R

and

| y = fCx^.x.^)}

by a 'source and target' picture:

R

R + (x„x2) f

f(x,,x2)

Figure 3 Now suppose we have two functions

f^, f?

of two variables

(x^,x2)

We can consider them both at the same time by writing

f(xisx2) = (f-L(x1,x2), f2(x1,x2))

so that

f

is a function from

R

2

to

R

e

R

X

R = R‘

2

The 'graph' picture in this case is harder to visualize, since by analogy with the previous definition of graph we have

2

graph(f) = {((Xl,x2),

(y15y2))

e R2 x R2

| y. = f.^^) i = 1,

and so the graph is a subset of

2}

R2 x R2 = R^

In a similar way, if we are given

k

functions of

n

variables we can

put them all together to obtain a corresponding function from We write

x= (x^x^

. . . ,x ) ,

,

Rn

to

Rk .

let

f(x) = (f1(x), f2(x),

fk(x))

,

and keep in mind the picture:

In formal notation this would be written as

f The graph of

f

: Rn - Rk

.

would be a subset of

R

n

x R

Tc

= R

n "hlc

t

difficult to

visualize and in general not yielding as much intuitive information about the behaviour of way

f

causes

f R

n

as we might extract from Figure 5 by considering the •

•

•

to be folded up and twisted inside

R

k

The techniques for analyzing such 'folds and twists' are those of topology and calculus together, or diffeventual topology.

Thus we see already how

a study of differential topology may give useful insight into the ways in which

k

functions of

n

variables can mutually interact both in general

3

circumstances and in particular cases.

Our aim in the first two chapters

will be to develop some of these basic techniques.

Remarks 1. It is frequently necessary to consider functions which are not defined for all values of the variables

(x^x^, .. . ,xn)

• belonging to some subset

x = (x^ jX^, • . . >xn)

but only for c say, of

U,

on K

In this case we of course write f : U -+ Rk

.

2. The word function is usually reserved for reaZ-vaZued functions only, i.e. those of the form f

In other cases

:

(something) -* R

(e.g. for

f : U

R

, k > 1)

we tend to use the term

map or mapping.

1.2

CONTINUITY

Roughly speaking, a map is continuous if by making a small perturbation in the input (i.e. the independent variables) you obtain only a small change in the output (i.e. the dependent variables). far too vague a definition. function

g : R -* R

However, this is obviously

For example, we would wish to think of the

defined by

f (x) =10

23

x

as being continuous

(indeed, its graph is a straight line), but it could be argued that small changes in

x

produce very large changes in

definition of continuity for functions

(a)

Continuity at a point

x

o

.

if, given any positive number

4

g(x).

f : R -* R

The function e

The formal is as follows:

f

is continuous at

x

o

(thought of as admissible margin of

error in the output), number

6

it is then possible to find another positive

such that perturbing

to vary by less than lxD - x| (b)

Continuity. R,

c.

< 6

Xq

by less than

|f(xq) - f(x)|

If we are considering

each point

xq

f

causes

f(x )

Symbolically,

implies

then we say that

6

f

U.

.

defined on some subset

is continuous on

belonging to

< e

If

U

U

U

of

if it is continuous at

is the whole of

R

any case understood from the context then we simply say that

or is in f

is

continuous.

Note that the function any

xq

and any

e

g

in the above example is continuous, since for

we may take

6 = 10

-23

e .

Remarks 1.

It is tempting to try to combine (a) and (b) by saying (hopefully)

f

is continuous on

U

if, given any

e > 0,

there exists a

that

6 > 0

such that |x — y|

for all

x

and

y

R

(b) but it does

The point is that for the

genuine definition of continuity we must allow well as on

< e

6

to depend on

xq

(as

If we do not do this, but demand that the

should apply everywhere (given

e),

then we have uniform

continuity - a notion which is in fact of considerable importance in contexts involving approximating functions by other functions as

for example in certain techniques of numerical analysis. 2. All

'standard' functions such as polynomials, exponentials,

sin x

and

so on can easily be proved to be continuous where they are defined. The only hazard to continuity in combining them ad libitum is the risk of dividing by a function which vanishes somewhere.

It is easy to see how the definition of continuity will go over to functions

f : R

n

R

k-

,

since all that is necessary is to replace the

modulus by the euclidean distance from the origin in Example 3 below).

Rn

or

(see

R

However, we shall need to study the 'small change in

input gives small change in output' problem in situations where the input and output may be rather more complicated than numbers or n-tuples of real numbers; for example,

they may be differential equations, or perhaps

collections of functions.

Therefore we want to generalize the definition

of continuity so that it applies to maps sets other than

Rn

or subsets of

distance in both sets f : A -> B 6 > 0

(b)

A, B;

f : A

.

Rn

6

than

e;

A

and

B

are

then we could simply say

is continuous at

x

e A

o

then the distance from

f : A ■+ B

where

To do this we need notions of

if, given

such that if the distance from

than

B

f(x)

x to

to

e X

o

f(xo)

> o,

there exists

(in

A)

is less

(in

B)

is less

is continuous if it is continuous at every point in

A.

Now it turns out that the minimal properties that a 'distance function' d

on a set

S

needs to have in order to reflect adequately the basic

relationships of distance in euclidean space are these:(1)

6

d(s,s')

is always

$ 0,

and

= 0

when and only when

s = s'

;

(2)

d(s, s') = d(s',s)

for all

s

(3)

d(s,s") £ d(s,s ' ) + d(s',s")

and

s'

for all

in

S ;

s, s'

and

s"

in

(This last property is known as the triangle 'inequality.) function satisfying (1),

(2) and (3)

S . A distance

is called a metric.

DEFINITION A function

d

which satisfies (1)3

S.

S

together with a particular metric on it is called a metric

A set

(2) and (3) above is called a metric on

space.

Note that function on

d S

is actually a function from itself.

The image of

d

S x S

to

and not a

R,

is contained in the set

non-negative real numbers (by (1)), and so we could write

d

R+

o

of

: S x S -> Rq .

EXAMPLES of metric spaces

.

1

d(x,y) =

S = R,

2. S = R2,

d(x,y) =

S = Rn,

d(x,y) =

|x -

(X1

n 3.

l i=l

Examples 1 and 2 are special cases of Example 3, known as the euclidean metric or usual metric in

4.

S = {bounded functions d(f,g)

5.

=

sup a $ x $ b

|f(x)

f

:

- g(x)|

R

[a,b]

-* R}

.

s = {differentiable functions

f

:

(a,b)

R

with bounded

derivative} d(f,g)

=

sup a 0

given any point

such that the halt

B (x)

x

in

W

is contained in

o

there W.

This leads to a general definition:

DEFINITION Let

S

he a metric space.

any

x

in

VI3

For example, line

R,

A subset

W

of

there exists some

6 > 0

the interval

2 < x < 3}

(x |

whereas the interval

(x |

S

with

2 < x $ 3}

is an open set if3 given B (x) o

{(x,y)

|

2 < x < 3, y = 0}

W.

is an open set in the real is not (because

fails to lie in any 6_ball contained in the interval). line segment

contained in

Observe that the

is not an open set in

since every 6-ball will now have to contain points with

x = 3

y f 0.

R2 , On the

other hand ((x,y)

|

2 0

we must show that Then

with

6

shown that

.

f ^(V)

6 > 0

is continuous.

f ^(V)

is open in

and so since

B (f(x)) cz. V . e

the existence of some B.(x) £2 f ^(V)

f(x) e V,

f

with

V

Let

V

A.

Choose any

is open we can find

Now the continuity of

f

guarantees

f(B^(x)) c B^(f(x)) c. V ,

This argument applies to each

be an

x e f ^(V)

so ,

so we have

is open.

Conversely, let us assume the property about open sets and deduce the continuity of e-ball

B^(f(x))

f.

Choose a point

is an open set in

B

x e A.

This means that since

x e f ^"(B (f(x))) e

B. (x)ct f ^(B (f(x))) o e

or, in other words,

This applies for each

e > 0 ,

Since

x

e > 0

the

(an easy exercise, again using

the triangle inequality), and so by hypothesis A.

For any

f ^(B^(f(x))) there is some

is open in 6 > 0

f(B.(x))cr B (f(x)) o e

.

and so proves the continuity of

was an arbitrary point of

A

we have shown that

f

with

f

at

x.

is

continuous.

In view of this theorem it is clear that in the study of continuity of maps between metric spaces it is the family of open sets in each space which is important, rather than the actual metric.

More precisely, if

two different metrics give rise to the same family of open sets then any map which is continuous using one metric will automatically be

12

continuous using the other. The family of open sets of a metric space is called its topology. All that is needed, therefore, in order to define the concept of continuity of maps from any set

A

they are given as metric spaces)

is the notion of a topology for each of

A

and

to another set

B

(whether or not

B.

Naively, we could select any family

F

postulate that

F

of subsets of

shall be the topology for

A.

A

and

Not surprisingly this

does not lead very far, unless we insist that the family

F

obey a few

simple rules which the family of open sets in a metric space does obey. The rules which turn out to be crucial are these: (1) If

U

and

(2) If each

IT

V

belong to

F F,

belongs to

then so does

U n V .

for some family

A

then so does

U U. X e A

{U } A X e A

.

For convenience we also add a formal clause: (3) Both the whole set to

A

and the empty set

0

belong

F.

Note that clause (1) number of sets in

F

implies that the intersection of any finite

also belongs to

F,

but it does not guarantee

anything about the intersection of an infinite number of them.

On the

other hand, clause (2) asserts that the union of an arbitrary number of sets in

F

must still belong to

F.

It is straightforward to verify (making good use of the triangle inequality) (1),

(2)

that the family of open sets in a metric space does satisfy

and (formally)

(3).

This means that we are now able to define

13

what we mean by a topology on an arbitrary set, whether or not we are given a metric.

DEFINITION

Let

be a set.

S

A family

F

of subsets satisfying the rules (1)3

and (3) above is called a topology for

S.

The set

(2)

together with

S

its topology is a topological space.

By analogy with the metric space situation, of

S

the sets in the topology

are referred to as open sets.

EXAMPLES of topological spaces 1.

Obviously,

every metric space is a topological space - since the

properties of open sets in a metric space inspired the general definition of

topology.

Rn

The topology arising from the usual metric on

is

called the usual topology. 2.

If

S

satisfies

is any set then the family of all subsets of (1),

(2)

discrete topology. discrete metric

and

(3)

S

and so is itself a topology.

naturally It is called the

It is the topology obtained by giving

(§1.2, Example 8),

S

the

as follows easily from the

observation that each point by itself forms an open set with respect to the discrete metric because when 3.

B^(x)

consists of

x

alone

6 < 1.

At the other extreme, we may take the family consisting of no open

sets except those legally required by clause 0.

4. Suppose

is a set with a given topology

S

arbitrary subset of

(3), namely

S

itself and

This is sometimes called the indiscrete topology.

the empty set

14

the point

S

F,

and

(not necessarily belonging to

T

F).

is some Then

T

can

be given a topology V

is of

the form

F,^

as follows:

T n U

The verification of

where

(1),

(2),

U

(3)

V

belongs

to

precisely when

is some set belonging to for

F^

is

easy.

This topology

is called the induced topology or subspace topology on from a metric on

S

(§1.2, Example 7)

on

5.

If

S, T

then

F

T.

where

U, V

unfortunately this fails if

S = T = R

to satisfy rule

S x T S, T

(2)

from those of

S

to be open if it is of

respectively, but

in general.

(For example

U x V

can be thought

and a union of rectangles may not be a rectangle.)

Instead we define a set in

of the form

are open in

S x T

with the usual topology then sets

of as rectangles,

comes

spaces there is a natural way of

tempting to define a set in

U x v

the form

F

If

T.

are two topological

It is

T.

F^

in fact comes from the induced metric

constructing a topology on the cartesian product and

F.

U x v

where

to be open if it is the union of sets

S x T U,

V

are open in

S, T

respectively.

It is

straightforward to verify that this does satisfy the rules for a topology on

S x T.

It is

called the product topology

on

S x T.

It is an instructive exercise to check that the product topology on Rn x Rm

(Rn,

Rm

on

R

= R

x R

6.

Let

S

with usual topology) .

be a topological space,

the disjoint union of a For example,

is the same as the usual topology

and suppose that

(possibly infinite)

family of

S

is expressed as sets

in certain types of application the elements of

. S

some objects which we are attempting to classify and each represent a collection of objects having a certain property

may PA

If we now broaden our focus and decide to regard two objects as if they belong to the same

S

,

may be

in common. 'the same'

we are led to consider instead of

S

a

15

new set

S,

This set

namely the set whose elements are themselves the

S

inherits a topology from that of W c. S

Given any set union of all in

S

if

those

W

let S

W

is open in

in the following way.

denote the subset of

which belong to

A

S

S.

W.

S

consisting of the

Then define

See Figure 7.

.

W

to be open

Again,

S

w Figure 7 it is routine to verify that this defines a topology on

S.

There is an equivalent but neater way to express this construction. Let

S

be a topological space and let

defined on S

S.

R

be some equivalence relation

The equivalence classes of

into disjoint subsets

S

as above,

R

give a decomposition of

and conversely such a

decomposition defines an equivalence relation on belonging to the same S/R:

S

.

this corresponds to tt

taking each be open if above,

A

x e S

it ^(U)

S/R (= S)

that of

We have a natural map

S -> S/R

to its equivalence class. is open in

and so it equips

- namely,

Denote the set of equivalence classes by S.

:

S

S/R

S.

Then this

with a topology.

Define a set is

U cz S/R

the same definition as This

topology on

is called the quotient topology.

If the topology on

S

comes from a metric it will not

necessarily follow that the quotient topology comes from a metric on

16

to

S/R.

there is no such thing as a

'quotient metric'

in general,

This is

one more reason for studying topologies rather than metrics.

Returning now to the motivating idea behind the generalization of metric space to topological

space, we will formulate a definition of

continuity for maps between topological spaces.

In view of the above

theorem and the subsequent discussion there is only one plausible definition that coincides with our previous definition for metric spaces.

DEFINITION

Let

f

:

undevstood). V

be a map between two topological spaces (topologies

A -* B

Then

is open in

f

is continuous if

f ^(V)

is open in

A

B.

Again we emphasize that there is nothing here to say that if open in

A

whenevev

then

f(U)

should be open in

U

is

B.

The main advantage of this topological definition of continuity is that it avoids reference to any specific metric, whether or not there is one available.

It is often useful to work with topological spaces

(such as

function spaces) which can be given metrics but only in rather artificial ways that distract attention from the more natural topological structures. Indeed,

there are contexts in which it is necessary to deal with

topological

spaces having no metric structure at all which is compatible

with the topology. The second advantage is the usefulness and economy of the definition for handling theoretical

statements about continuous maps.

For example,

consider the following proof of the fact that the composition of two continuous maps is continuous. is of course straightforward,

The proof for metric spaces

(using

6,

e)

but not as tidy.

17

THEOREM

If

f

A -* B

g*f

: A

:

C

and

g

:

are continuous then so is the composition

B

C

in

C.

.

Proof Choose any open set continuity of

g,

V so

f ^(g ^(V))

f.

But

1.4

FURTHER TOPOLOGICAL CONCEPTS

(i)

f ^(g 1(V))

Then

An open set

neighbourhood of

U

=

(g*f)

x.

of

*,

A

by the

by the continuity of

this finishes

x

B

the proof.

is called an open

For technical reasons it can be useful to think in

x

and to call any set

N

simply a

if it contains an open neighbourhood of

Thus for example the interval (though not open)

is open in

is open in

containing the point

slightly more general terms,

neighbourhood

^(V)

g ^(V)

(x e R

of the point

of either of the points

1

or

|

1 $ x $ 3}

2

in

R,

3

in

R.

x.

is a neighbourhood

but it is not a neighbourhood

There is a characterization of open sets that is convenient to use in practice when checking whether a given set

is open or not.

straightforward from the definition of topology

The proof is

(page 14) with clause

(2)

playing an important part.

Characterization of open sets A set

U

is open if and only if for each point

neighbourhood

N

of

X

x

with

N

X

We see from this that we can use in contexts where we would use can define a map

18

f

:

A -> B

B^

contained in

x e U

there is a

U.

'neighbourhood'

in topological

in metric spaces.

to be continuous at a point

spaces

For example, we x

e A •

(a concept which we have only used so far in metric spaces)

neighbourhood

N

of the point

yQ = f(xQ)

neighbourhood

M

of

f(M) B

page

with

xq

e B

between two topological if and only if it

x^ e A.

if given any

that a map

spaces is continuous

is continuous at

(as defined on

(as defined above)

for

This now brings us back satisfyingly in a full circle

to the original definition of continuity in a metric space

(page

9) with

which we began.

(ii)

Given any set

W

in a topological space

union of alt the subsets of

W

we can consider the

S

which are open sets in

S

with the induced topology - but the converse may not hold). open

(by clause

contained in

It is called the interior of

measures how 'thick'

W

has empty interior in in

R

is

W

This is

and so can be thought of as the largest open set

(2))

W.

(so open in

is as a subset of R

.

the open interval

S.

W,

and in a sense

Thus a line or a plane

The interior of the closed interval

[a,bj

(a,b), but both have empty interior in the

2 plane

R .

The set

Q

of rational numbers and its

set of irrational numbers,

(iii)

open nor closed; simultaneously. open and closed. open.

each have empty interior in

A subset of a topological

complement is an open set.

complement I,

the

R.

space is called a closed set if its

Generally speaking, most sets are neither

it is also possible for a set to be both open and closed In a discrete space,

for example,

all

sets are both

Thus a closed set is not simply one which fails to be

The reason for introducing this

terminology is that topological

arguments frequently go through much more rapidly when expressed in terms

19

of closed sets

than they would do if they dealt with open sets directly.

Observe that by taking complements

the useful

characterization of open

sets given above is transformed automatically into the following:

Characterization of closed sets A set

C

is closed if and only if for each point

a neighbourhood

(iv)

Let

point

x

W

of

entirely disjoint from

(which may or may not belong to

apart from (possibly) the point

W,

points of

W

'arbitrarily close'

to

x

C

there is

S,

and define a

to be an accumulation point

W)

if each neighbourhood of

W

not in C.

be any subset of the topological space

or limit point of

point of

y

y

contains at least one

x

itself.

x

Thus there are

but different from

x.

It is

easy to see that the characterization of closed sets above can now be converted into the statement that a set is closed if and only if it

contains all its limit points. Given an arbitrary set manufacture a set points.

W

This set

and the set

C

W,

not necessarily closed or open, we can

consisting of

W

together with all

is called the closure of

W

is closed if and only if

C = C

W. .

its

limit

Obviously

W et W,

The following facts

are not difficult to verify: (1)

For any set

(2) W (3)

w2 = W

However,

=

20

the closure

W

is a closed set,

is the intersection of all closed sets

w1 u

= {x

W

u w2

note that e

R

|

r\ W2 = 0

x < a} but

i.e.

W = W

that contain

.

W.

.

Wj n ,

W2 1

n W

W2 = {x e R A W2 = {a}

.

necessarily. |

x > a}

.

Then

For example, A W2

take

If

A

is a subset of

equal to or contains

B,

B

and yet occupies so much of

then we say that

B

is dense in

A

that

A

or is

an accumulation point of

subset of

B

contains points of

has empty interior. (and open) in

R.

in

For example,

R.

The concept of denseness

dynamical systems, See §3.3

'usual'

or that the complement of

(Sard's theorem)

is

A

in

B

the set of non-integer numbers is dense (and not open)

is very important in mathematical

as opposed to

and so on:

B

or that every open

The set of rational numbers is dense

characterizations of

(v)

A,

A,

is

B.

Other equivalent ways of saying this are that every point of either in

A

'unusual'

properties of maps,

we shall make considerable use of it.

and also

§4.4.

The familiar ideas of approaching a limit and convergence can easily

be fitted into the topological framework constructed so far. space context f(x) any

(e.g.

in euclidean space) we say

approaches e > 0,

x ) o

< 6

as

c

approaches

x

there then exists some

property that d(x,

In a metric

d(f(x),

c)

< e

xq

6 > 0

for all

x

if,

given

with the with

.

In topological terms this becomes f(x) approaches any neighbourhood neighbourhood f(x)

e N

as

c

M

for all

approaches

x

xq

if,

given

N

of

c

there then exists some

of

xq

with the property that

x e M .

We write f (x) -> c

as

x -* xq

or

21

lim x

and call

c

->

the limit of

f(x)

X

= c

o

f (x)

as

x -> x

o

The wording of these definitions sounds familiar, that it is an echo of the definition of earlier in this section. these ideas?

What,

then,

and in fact we note

'continuity at a point'

given

is the exact relationship between

Proceeding carefully from the definitions alone we soon see

that to say f(x) -> c

as

is the same as to say that the map g(x)

= f(x)

x -> x g

defined by x i x

when

o

o •

g(xo) = C

is continuous at a limit'

x

.

Therefore in making the definition of

'approaching

we have actually introduced nothing new.

A standard variant on the above occurs when we are dealing with a

sequence of points

x^, x£, x^, x

n

-> c

... as

.

In this case we say

n -> °°

or lim x = c n n -> °° to mean that given any neighbourhood

N

exists some positive integer x

n

e N

for all

For metric spaces usual

22

of

c m

there then such that

n > m .

(in particular euclidean space)

idea of a sequence approaching a limit:

this coincides with the

given any

e > 0

there

exists some positive integer

m

such that

d(x n’

all

c) '

is less than

e

for

n > m .

Here,

and in the general

is convergent,

topological setting, we say that the sequence

or converges to the limit

In the special

c.

case of real or complex-valued functions or sequences

the definitions can be stretched a little to include the notion of ’approaching infinity’. set of the form (or

xn -*■ + °°)

{y e R

|

We replace the neighbourhood

{y e R

|

y > r}

for some

in the real-valued case,

y < s}

for some

s

r

N

when defining

or by a set of

when defining

of

c

by a

f(x) -> + °o

the form

f(x) -> - °°

(or

x

-> - «>) n

also

in the real

{zeC|

| z|

case,

> r}

or,

finally,

to define

by a set of the form

f (x) -> °°

(or

-> °°)

in the complex¬

valued case.

Remarks 1.

The definitions for sequences can actually be subsumed under those

for functions.

The way to do this is to consider the sequence

x^,

of points

1,

X£, x^, 2,

3,

...

...

and so on. to

N

in the space

under the map

c

as

n

°°’

(n) -> c We will not go into details, There is an irritating

this point.

as the images of the integers

taking p

1

to

x^,

(thought of as

N u {p}

and putting a suitable topology on

re-express

2.

:

S

2

to

x^,

’infinity’)

it is possible to

as as

n

p

.

since they are rather uninstructive.

source of ambiguity which inevitably arises at

We have the notion of 'limit of a sequence and limit point

of a set, but when we think of a sequence as a set of points these two

23

notions may not coincide. -

I +2 2’

1_

_5

4’

6’

i.e.

8’

although the points On the other hand,

For example,

±1

x

n

=

(-1)

the sequence n 2n - 1

has no limit,

2n

are both limit points of the set

the sequence

1,

1,

1,

1,

...

(x^

n e N}

|

has the limit

1

.

but

as a set it has no limit points since it consists of only one point. Therefore care must be taken to use these terms with precision when sequences are being treated from a topological point of view. 3. When

S

is a topological space which also carries some notion of

addition (for example a normed linear space:

see §2.3),

then besides

the

concept of convergent sequence we also have that of convergent series. A series means simply a sequence of elements of

S

with

'+'

signs

between them:

x^ + x^ + x^ + ... We say the series is convergent or otherwise according to whether the sequence

Sf, $2,

, ... are the so-called -partial sums

is convergent or otherwise, where the n

s

We emphasize, however,

n

=

y x. .r i=l

.

that this only makes sense to the extent that 00

'+' makes

sense in

S.

4. Any metric space

given

x,y e S

such that

U

S

We write (such as

24

Rn)

and

V

are disjoint.

where

d = d(x,y)

to denote

U, V

of

x,

respectively

y

This may seem obvious,

In fact we may take :

lim S n->-°°

has the elementary property that

there exist neighbourhoods

verified from the definit'ons. V = Bd/2(y)

Y x. i=0 1

U = B

but must be

d/2 then the non-intersection of

(x)

and

U, V

follows from the triangle inequality.

Unfortunately topological spaces

failing to have this

'separation'

considerations

in quotient spaces:

(e.g.

property can arise from quite natural see page 16,

and §4.1).

Spaces which do have the property are called Hausdorff spaces (F.

Hausdorff,

1868-1942).

Thus all metric spaces are Hausdorff.

1.5

HOMEOMORPHISM OF SPACES AND EQUIVALENCE OF MAPS

Let

S, T

be two topological spaces,

and suppose

f

:

S -> T

is a

*

bisection f

:

.

If

T -*■ S

f

is continuous,

is continuous,

definition of continuity,

concerned

S

f

:

and

S -> T T

f

called a homeomorphism.

is

this means that the open sets in

precisely to the open sets homeomorphism

then

and at the same time its inverse

in

T,

under

f.

Thus

then as far as their topological structure is

are indistinguishable.

any property possessed by S

We say that

S

will also be possessed by

T,

f

^

is continuous:

section for a particular theorem on these lines. then not necessary to study

f ^

T

are

It follows that

for example)

f

is continuous

see the following This means that it is

explicitly in order to show that

a homeomorphism, which is very useful

generality.

and

and vice versa.

In many useful contexts the fact that a bijection guarantees that its inverse

S

and expressible purely in terms of the

(such as the property of being Hausdorff,

difficult to analyze.

correspond

if there exists a

topologically equivalent or, more usually, homeomorphic.

topology of

S

From the

since

f ^

f

is

may in practice be

Unfortunately this result does not hold in all

It may fail,

for example,

in certain function spaces.

Artificial counterexamples are also easily manufactured. with the discrete topology and

T =

R

Let

with the usual topology,

S =

R

and let

* see page 271. 25

f

:

S -> T

domain U

S

in

S

be any bijection: is continuous!) with

fU

then

but

f

f ^

not open in

is continuous

(every map with

is not since there exist open sets

T.

Homeomorphism is the natural notion of equivalence between two topological spaces.

We next consider how to define a notion of

equivalence between two maps S, T

are given.

f,g

:

S -* T

Thinking informally of a homeomorphism from a space

to itself as the result of viewing we can regard two maps, like

f

when

S

and

f,g T

:

S

S T

through distorting spectacles, as being equivalent if

g

looks

Formally this

is expressed by saying that

are topologically equivalent or conjugate if there exist homeomorphisms

g

h

:

S

S

k

: T -> T

such that g-h = k*f i.e.

the result of applying

g

same as the result of applying by

k.

to f

S to

, after distortion by S

h

is the

and then distorting the result

Note that this is the same as g = k*f*h 1

,

and can be expressed simply by saying that the following diagram f S h

->

j. S

T |

g -*

k

T

commutes, i.e. either pathway from the top left hand corner to the bottom right hand corner gives the same result.

26

S

are both viewed through distorting spectacles -

but with separate lenses. f,

where the two topological spaces

Observe that since

h, k

have inverses

the relation of topological equivalence is an

equivalence relation on the set of continuous maps In cases when h,

S = T

we often insist that

k

S -> T. should

since we wish to avoid the idea of distorting

ways

simultaneously.

S

he the same as

in two different

The expression for topological equivalence then

becomes g = h*f*h 1 We shall see more of this later, when studying the qualitative theory of dynamical

1.6

systems

(§§4.2,

4.5).

COMPACTNESS

There

is an extremely useful theorem in analysis as follows:

THEOREM A

If

is a continuous real-valued function on a closed interval

f

then

f

| f (x) |

is bounded, £ k

for all

i.e. x

there exists a constant

in

r

tt

[0, -j)

tan x

such that

[a,b] .

Note that the assumption that For example,

k

[a,b]

is a closed interval is vital.

[a,b]

is defined and continuous on the non-closed interval

but is certainly not bounded.

Here are two standard proofs of the theorem.

Proof 1 Suppose exist

f x^ e

is not bounded: [a,b]

with

|f(x^)|

Then there exists

x2 ^ x^^

by the greater of

2

points

x-j-,

x2,

x3,

and ...

then we deduce a contradiction. > 1

with ^(x^l

with

(otherwise

|f(x2)| ).

lf(xn)|

> 2

f

There must

is bounded by

(otherwise

f

1).

is bounded

Continuing, we obtain distinct > n

for each positive integer

n.

27

Now we invoke the

Weierstrass limit point theorem (or

'classical'

condensation theorem):

If

W

A

is a subset of a closed interval and

contains ccn infinite

A

number of points then there is at least one accumulation point of

A

somewhere in the interval.

This immediately implies that there is in point x^.

xA

A = {x^, X2, x^,

of

But by the continuity of

there is a 5^ > 0 B

(x^).

such that

Choose now

m

...},

i.e.

f

x^

at

|f(x)|


n £ m >

|f(xA)|

(Note:

we do not claim that

x

+ 1

avoids

l

with

n

x,, x„,

*

n :> m,

x

z

n

.

m-1

and

which gives the contradiction. x

-> x. *

.)

By the continuity of

f

n

Proof 2 Choose [a,b]

e > 0.

there exists

provided

y

6^ > 0

is within

the open intervals

6^

such that of

(x - 6

x.

x + 6 )

we know that for each |f(y)|

is less than

The whole of as

x

[a,bj

x

in

|f(x)|

+ e

is covered by

runs through

fa,b1 ,

or

equivalently [a,b]

=

U x e

where the

'

(x - 6

, x + 6 ) '

[a,b]

denotes that part of

(x - 6

, x + 6 X

(Note in passing that

(x - 6

the induced topology of the Heine-Borel theorem:

28

[a,b]

, x + 6 )'

) X

lying in

is an open subset of

as a subspace of

R.)

This

[a,b]. [a,b]

in

time we use

Suppose

HB

-is a closed interval in

Ea>tT|

there is given a family [a,b] c

U J

(with

R

of open intervals covering

{J^}

Then there is a finite sub-family

.

\

J

A

which also covers

Applying this

[a,b] ,

i.e.

Qa»bJ cz J

than

[a,b] c= J. v X -i

to the family

deduce that there exist u J u X0 12

X-.

...

{J

x2, u J

X

a,b / ± «>)

x

}

where

..., xm

.

J,

u

x

in

=

, J , . .., J X, X0 A 12m ...

uJ X

(x - 6

[a,b]

Then clearly

i.e.

[a,b] ,

An

J

ccnd

x

, x + 6 ) x

we

with

If(x)I

is everywhere less

m

max{ | f (x-^) | , | f (x^) | ,

...»

| f (x ) | } + e

which shows that

f

is

bounded. It is

straightforward to follow through the above two proofs and check

that they work for functions defined on any topological space satisfies an analogue of interval

[a,b2

open set of

S

W

or

is replaced by in

HB

HB S

S

which

respectively where the closed

and open interval is replaced by

In fact topological spaces that satisfy this

analogue of the Heine-Borel theorem play such a fundamental role in analysis that they are given an explicit name:

compact spaces.

DEFINITION

A topological space

S

is compact if and only if every cover of

S

by

open sets has a finite subcover.

Sometimes it is convenient to talk about a compact subset space

S.

This

simply means that

topological space in its own right

T

of a

is compact when regarded as a

(with the induced topology).

In the present more general context it is easy to deduce compact spaces as a consequence of

T

HB

.

W

for

The argument goes as follows.

29

Suppose

A

is a subset of the compact topological space

an infinite number of points. point.

If not,

We wish to prove

then every point of

containing at most one point of

A.

S

A

S

and

A

has

has an accumulation

has an open neighbourhood

But then the compactness of

S

implies that a finite number of these open neighbourhoods will suffice to cover

S,

which means that

This contradiction shows that

A

can have only a finite number of points. A

must after all have an accumulation

point. Returning to the starting-point of this section, by mimicking either Proof 1

(using the above generalized version of

W

)

or Proof 2

directly we can construct a proof of a more general version of the theorem on boundedness of continuous functions.

THEOREM A'

If f

is a compact topological space and

S

is bounded,, i.e.

for all

f

there exists a constant

: k

is continuous then

S -*■ R

such that

|f(x)|

$ k

x e S.

Remarks 1. Unfortunately it is not possible to deduce compactness from the Weierstrass property in the case of an otherwise arbitrary topological space.

Thus the generalized versions of

equivalent.

i.e.

2. Theorem A' ways

HB

are not entirely

they can be shown to

In particular they are equivalent in their original for closed intervals

[a,b]

of the real

line.

has an immediate corollary which illustrates one of

the main

in which the theorem might be used in practice, namely that if

is compact and

30

and

In the case of metric spaces, however,

be equivalent. context,

W

f

:

S

R

S

is a continuous non-vanishing function then

f

is bounded away from zero, |f(x)|

^ k

for all

i.e.

x e S.

there exists

k > 0

such that

The proof of the corollary comes from

applying Theorem A

to

1/f.

Clearly this need not be true if

not compact:

S =

(0,1)

and

take

In applications

f

f(x)

= x,

S

is

for example.

may typically be a distance function from points of

S

to some other point or set.

3.

There is an alternative characterization of compactness as follows.

A topological

space

closed subsets

is compact if and only if, given any family of

S

such that any finite number of the

{C^}

non-empty intersection, At first sight this

have

then the intersection of them all is non-empty.

seems unrelated to the former definition,

but in fact

it is easy to translate one version into the other simply by considering the open complements

U

A

of the closed sets

Thus for example the statement

S = U U A

C. , A

becomes

and vice versa. 0 = f| C

A

A

,

and so on.

A

We leave this verification as an exercise. This characterization is frequently used to prove the existence of a point defined by some kind of infinite intersection process. illustration,

observe that if

I

is

the closed interval

the intersection of any finite number of

I

's

As an

To, — 1

then

is non-empty and in fact

CO

f]

I

= {0}

which is non-empty.

On the other hand if

=

(0, — ]

n=l

then again the intersection of any finite number of

J^'s

is non-empty

00

but now

0

J

= 0.

Here

=

(0,l]

is not compact

(see below).

n=l We will now give a working theorem whose geometrical usefulness justifies

the rather technical definition of compactness.

three facts about compact sets,

First we state

each of which is straightforward to verify

from the definitions:

31

(1)

Every closed subset of a compact space is compact.

(2)

If

f

:

S -* T

compact subset, (3)

K c. S

is continuous and then

f(K)

is a

is a compact subset of

T.

In a Hausdorff space every compact set is closed.

Putting these together we obtain

THEOREM B

Let

and

S

Suppose

f

be topological spaces with

T :

compact and

S

T

Hausdorff.

is a continuous map which is also a bisection.

S -> T

is automatically a homeomorphism (i.e.

Then

f

is automatically continuous).

f ^

Proof To prove

f ^

is,

V

i.e.

bijection) implies

continuous we have to show open implies

as showing

C

compact

C

(Fact

which in turn implies

f(V)

open.

(f

f(C)

(1)) which implies closed

is open whenever

This is the same

closed implies

f(C)

^(V)

(Fact

closed. f(C)

V

(since

f

But

closed

compact

C

(Fact

is a

(2))

(3)).

Thus for example to prove that the space of real numbers modulo the integers

(i.e.

the quotient space

R/R

where

xRy

means

x - y e Z)

is

formally homeomorphic to a circle it suffices to construct a continuous bijection R/R

from

is compact

and of

f

R/R

(being the image of

is Hausdorff. 0) h- e^771®

1 :

0 H- e

2 tt

of working with

32

i0

f

R/R

.

[0,l]

in the complex plane, under the projection

It is easy to verify that

is a bijection,

quotient topology in R -> S

to the unit circle

f

:

R

since R/R)

(equivalence class

and is continuous by definition of the

and the continuity of

the map

We have no need to bother with the technicalities .

Recognizing compact sets.

In all the above theory we have not discussed

how to recognize a compact set where we see one.

After all,

it is

clearly impossible to check the properties of every open cover. Fortunately,

in euclidean space

Rn

,

the most useful case,

there is an

immediate test:

THEOREM C A subset

of

K

is compact precisely when it is both (a) closed and

Rn

(b) bounded (i.e.

there exists some number

The compactness of (cover

K

by

K

K fi B^(0) ,

k

immediately implies k = 1,

2,

3,

...).

with

(a)

Kc B^CO)).

(Fact

However,

more delicate, being a generalized Heine-Borel theorem. the proof here.

of

1.7

and

(b)

the converse is We will not give

Note that once we have Theorem C then the original

Theorem A follows directly from Fact bounded on

(3))

[a,b]

is the same as

(2),

saying

since to say that f([a,bj)

f

is

is a bounded subset

R.

CONNECTEDNESS

A topological space is connected if, piece.

Of course,

roughly speaking,

it is all in one

any space with more than one point in it can be

written as the union of two disjoint non-empty subsets, but it turns out that the definition which best captures our intuitive notion of being 'in one piece'

is the following:

DEFINITION

A topological space

S

is connected if it cannot be split into two

disjoint non-empty subsets which cere both open3 i.e. if U, V

S = U u V

where

are open and disjoint then one of them must be empty.

33

Thus in a connected space the only subsets which are both open and closed simultaneously are the empty set and the whole space.

It follows

that any space with the discrete topology cannot be connected if it has more than one point.

A subset

Q

of

Rn,

when given the induced

topology, will fail to be connected precisely when it can be written as Q = Qx U Q2

where

some open set

Ik

in an open set

Q-^,Q2 of

Ik

the notion of

are non-empty,

Rn

disjoint,

'separate pieces'.

of the two closed intervals (as a subset of

R

Q1 = Q A (-2,2)

and

Q.

- in other words when each

which does not meet the other being

and

=

[o,l]

and

with the induced topology), Q2 = Q

for

can be included

Q_^.

This formalizes

For example, Q2 =

= Q n U.

the union

Q

is not connected

[2,3]

since we have

C\ (1,4).

This still leaves the problem of deciding when a given topological space -is connected,

since it is impossible to check all possible attempts

at splitting it into disjoint open subsets.

The two most useful results

in this direction are:

(1)

If

S

then

is connected and f(S)

f

:

S

T

is continuous

(with the topology induced from

T)

is

connected; (2)

every open, half-open or closed interval infinite)

(possibly

of the real line is connected.

Sketch proofs (1)

If

S = f

f(S)

= U u V

X(U) u f ^(V),

clearly disjoint.

34

(both open and disjoint in both open in

S

f (S))

then

by the continuity of

f

and

(2)

If an interval

in

I,

a < b,

numbers

x

X)

is not connected we have

disjoint and non-empty.

suppose

m

I

shows or

for which

m V

and let

b

in

m = least upper bound of the set

X

of all

U

Choose points

I

a

is contained in

in

both open

U,

[a,x) n

cannot lie in

I = U u V,

U.

Contemplation of

(since it would not be an upper bound for

(since it would not be the least upper bound for

Contradiction,

so

I

V,

X).

is connected.

As with compactness, we extend the definition to subsets of a topological

space by regarding them as spaces in their own right with the

induced topology. Apart from its direct geometrical meaning,

connectedness plays an

important indirect role in the proofs of many theorems. to prove that a certain property holds for all space

S.

the set S,

x

in some topological

A frequently useful approach is to prove first of all T

of points

x

for which

P

P

fails

is also an open subset of

the existence of just one point non-empty,

so

F

guaranteed for all

x

in

Remarks on the literature. [99] .

for which

If P

of points for S

is connected,

holds means that

T

is

is

There are many good introductory texts on such as Mendelson

[80] , Patterson

Particularly recommended also are Sutherland

and Simmons

P

then

S.

is written in much the same spirit as provided,

S.

F

must be empty and therefore the validity of

metric and topological spaces, Pitts

x

that

does hold is an open subset of

and then to prove by other means that the set

which

Suppose we wish

this

[94],

[L34] , which

chapter but with all details

[ll| , which contains much additional material

relevant to the notes as a whole. See Willard [l50] for a deeper study of topological spaces, or Hocking and Young [57] for more geometric aspects.

35

2 Calculus

2.1

DIFFERENTIATION

Recall from elementary calculus the definition of differentiability at a point

x

for a real-valued function of one real variable.

DEFINITION

The function

f

:

R

is differentiable at

R

lim

(f(x+h)

x

if

- f(x))

h - o h

exists (± °°

not admitted).

This limit is called the derivative of denoted by

f'(x)

or

f

at

x,

and is traditionally

df — .

Strictly speaking we should write

x

for which the limit has been evaluated.

to emphasize the point

Another way of expressing the existence of the limit is

there exists a real number f(x+h)

where In general.

X

e -> 0

as

- f(x)

X

h -> 0

36

f(h)

= f(x+h)

L(h)

= Ah :

to say:

= Ah + e|h|

of course depends on

h = 0

,

such that

x.

that the graphs of the two functions

are tangent at

df -j^- (x)

- f(x)

see Figure 8.

f, L

The pictorial of

h

interpretation is

defined by

It is reasonable to describe this by saying that

approximation to

f

at

x, h

or, f (x)

equivalently,

L

is a 1■■inear

that the map

+ Ah

(whose graph is a straight line)

is a first order approximation to the

map h

f (x+h)

Since this idea turns out to be so useful in the study of real functions of a real variable, much as possible. (maps)

it is worthwhile trying to generalize it as

Obviously we would get nowhere by considering functions

between arbitrary topological spaces,

since the minimum necessary

ingredients are: 1.

concepts of addition and subtraction,

2.

the idea of a map being linear,

3.

a possibility of taking limits.

Now the first two of these are available in the world of linear spaces or vector spaces, while the third is most conveniently associated with metric spaces

(§1.2).

Therefore to generalize our simple notion of

differentiation we need to develop a theory of linear spaces which are at the same time metric spaces.

This will occupy us for the next two

37

sections of the chapter.

2.2

LINEAR SPACES AND LINEAR MAPS

A real linear space or vector space is a set

V

together with two

operations called 'addition' and 'multiplication by scalars (i.e. real numbers)' which obey a number of reasonable rules.

The standard rules

for addition are these: (i)

x + (y+z) = (x+y) + z

for all

x, y, z

(ii)

x + y = y + x

for all

x, y

(iii)

there is a zero in satisfies

(iv)

every

V,

in

for every

has a negative, i.e. an element

which satisfies

V

V

i.e. an element

0+x=x+0=x

x

in

0

which

x

in

V

- x

x + (~x) = 0 ,

followed by rules involving scalar multiplication as well as addition: (v)

a(x+y)

= ax + ay

for all

X,

y

(vi)

(a+g)x = ax + Bx

for all

X

in

V

and all scalars

for all

X

in

V

and all scalars

for all

X

in

V.

(vii) (viii)

a(gx)

=

(ag)x

lx = X

in

V

and all seal

Remarks 1. It can be deduced from these rules that other obvious-looking arithmetical statements also hold, such as Also,

it is usual to write

x - y

Ox = 0

instead of

or

x + (~y)

(-l)x = - x . .

2. The same definition could equally well be made with complex numbers rather than real numbers as scalars, giving a complex linear space. 3. When thinking geometrically we call

38

0

the origin of

V.

4. Nothing has been said about any topology for

V.

EXAMPLES of linear spaces 1.

V = R

2*

V =

3.

V = {all

R

;

usual addition and multiplication. ;

co—ordinate-wise addition and

R

functions

R}

defined pointwise,

;

scalar multiplication.

addition and scalar multiplication

i.e.

f + g

means the function

x

f (x)

af

means the function

x W

a,

g.

on the left hand side denotes addition in

whereas on the right hand side it denotes addition in completely different in the way they are defined.

40

(in

V)

to zero

(in

These may be

A similar remark

applies to the scalar multiplications on both sides. linear map always takes zero

W.

W).

V,

Observe that a

EXAMPLES of linear maps 1.

V = any linear space;

L

some fixed scalar R

,

W-R

;

: V -> V

defined by

j

L

•••)

n

:

V

W

(1 $ i £ m,

for

defined by n

n

)

“13

j‘i

for a collection of a£j

= av

a.

( n ^>^2

L(v)

mn

1 $ j

x3’

X

“2j v

l

•••

a

j=i

.

mj

x.

J

fixed scalars

£ n)

.

Here

L

is traditionally

represented by the matrix

all

a12

a21

a22

a2n

•

•

•

•

•

•

•

•

•

Otrt

^ ml

3. V = S L 4.

defined by

Linear maps from

(i)

L(f)

V

ot

mn^

above,

= f'

to

“in

•••

m2

in Example 7(b)

: V -> W

•••

W

W = V

in Example 3 above;

.

can be added together and multiplied by

scalars pointwise to give further linear maps. check that the set space,

Lin(V,W)

of linear maps

Indeed, V -> W

it is easy to

itself forms a linear

in fact a linear subspace of the linear space of all maps

V -* W

.

This construction is so important in what follows that we will isolate it for convenient reference:

of all linear maps

L

:

:

V -* W

If

V -> W

V

and

W

are linear spaces3 then the set

forms a linear space3 denoted by

Lin(V,W)

.

Kernel and image For a linear map elements

v

in

L V

let the kernel of

L

denote the set of all

which are taken to the zero element in

W.

41

Linearity of

L

implies that the kernel

V,

L

is injective precisely when

and that

Thus the

'size'

of

ker(L)

(for example,

measures the failure of injectivity of The image

im(L)

L

:

V

W

im(L)

its dimension:

W,

is all of

see below)

W.

inverse

L

: W -> V

must also

recall

§1.5 and Theorem B of

A bijective linear map is called a linear isomorphism,

L.

L

(Compare this with the analogous situation for

isomorphism the linear structure of via

V.

and by definition

isomorphism if the linear context is understood.

W,

in

isa linear map and is at the same time a bisection then

continuous maps between topological spaces; §1.6.)

= {0}

L.

it is a simple exercise to check that its be a linear map.

is a linear subspace of

ker(L)

is a linear subspace of

is surjective precisely when If

ker(L)

V

If

L

or just

: V -> W

is an

corresponds precisely to that of

Thus if there exists an isomorphism between two given linear

spaces they are indistinguishable as linear spaces:

they are said to be

isomorphic.

EXAMPLE Suppose

S, T

S n T = {0}.

are two linear subspaces of Then it turns out that every

a unique way as as

x = s + t

'co-ordinates'

for

x,

where

s e S,

S

and

such that x

t e T.

in

V

S + T = V

S x T.

We can regard

We write

and

can be written in

and it is routine to check that

isomorphic to the cartesian product

direct sum of

V

V

s,

t

is

V = S @ T,

the

T.

Dimension In euclidean space geometrical problems can be converted into algebraic problems by taking co-ordinates,

42

sometimes already given and sometimes

constructed artifically with the solution of a particular problem in mind. This

is

such a useful technique that it is worth exploiting as much as

possible.

It leads

to the concepts of basis and dimension in general

linear space theory. A linear space finite collection each

x

in

V

is said to be finite—dimensional if it contains a

V

(e^,

•••>

effl}

can be written in the form x - a,e1 + a„e0 + it 2 Z

for some unique choice of scalars depends on for

V,

of elements with the property that

x).

+ a e mm

a,,

cu, Z

...,

a

{e,,

eOJ

....

e }

1

The collection

and the number

...

12

(which of course

m

m

is called a basis

is called the dimension of

m

V.

It can be

shown to be independent of the actual basis considered. A linear space which is not finite dimensional is

dimensional .

Note that this

'infinite basis'

for

V,

said to he infinite-

says nothing about the existence of any

whatever that may mean.

EXAMPLES 1.

V = Rn

(0,

has a basis

I,

0,

elsewhere.

This

dimension of 2.

Let

W

is

V

...,

R

is is

{e^,

0)

with

n

in the

where ith

e^

is the vector

place and Hence the

be the linear space consisting of all maps

some given linear space,

For example, when

0

(as we would hope.).

V

dimensional precisely when

V

1

en^

called the standard basis.

scalar multiplication in

get.

•••»

A

is an arbitrary set,

are defined pointwise. W

W = R

A -> W

and addition and

Then

finite-dimensional and

is

and

can be thought of as an n-tuple

A= {1,

(x^,

2,

...,

•••> xn)

where

n}

V A

is finite¬ is a finite

an element of

with each

x^

in

R,

43

and

V

is

Rn.

isomorphic to

element of

V

R,

is infinite-dimensional.

and

V

A = N = {1, 2, 3,

If

is an infinite sequence

(x^,

*2, x^,

...}

. ..)

then an of elements of

3. Any linear space isomorphic to a finite-dimensional space is itself finite-dimensional, to a basis

since the isomorphism transports a basis in one space

in the other.

Thus any infinite-dimensional space cannot be isomorphic to a finite¬ dimensional one.

On the other hand,

of the same dimension are isomorphic. that every finite-dimensional

isomorphic to

Rn

any two finite-dimensional spaces From this and Example 1 it follows

(real) linear space of dimension

n

is

- although in practice the construction of an

isomorphism may be rather artificial.

After this excursion into the theory of linear spaces and linear maps we now turn again to the task of generalizing the definition of differentiability. If linear)

V, W

are linear spaces and

then given

as an element of

x

W.

and

h

F(h)

= f(x+h)

V

: V -*■ W

is any map

(not necessarily

we can certainly define - f(x)

Now if we wish to discuss approximating

some suitable linear map topological structure. differentiability at

in

f

L

F

by

we need to introduce some metric or at least

Although we could make a definition of x

involving only topologies on

V

and

W,

experience shows that in the first instance it is useful to ensure that V

and

W

are each metric spaces

(as well as linear spaces),

and moreover

that the metrics are in a certain sense compatible with the linear

44

Pursuing these ideas we arrive at the notion of a normed

structures.

linear space.

2.3

NORMED LINEAR SPACES

Observe that (i)

(ii)

in

R

with the euclidean metric the following rules hold:

d(x+z, y+z)

= d (x, y)

for all

x, y,

z

in

Rn

(i.e.

translation does not alter relative distances).

d(ax,

ay)

scalars

=

|a|d(x,y)

a

(i.e.

for all

x, y

in

Rn

and all

scalar multiplication magnifies or

contracts distances uniformly). These are not consequences of the general definition of metric since they involve the linear structure of

Rn

(§1.2),

which need not exist in

an arbitrary metric space. Let us d

say (temporarily)

on it then

that if

x,

length of

the

(1)

y

in

=

and then since (2)

||x|| when

(3)

V.

|a|

d

||x||

and

(ii).

d(x,0)

by

||x||

we see that in view of for every

x

in

V

If

d(x,y)

d

is

= d(x-y, 0)

(thought of as the (ii)

above we have

and scalar

a,

is a metric we also have

£ 0 x

||x+y||

Ox )

(i)

we immediately see that

Denoting

'vector'

||ax||

is a linear space with some metric

is sensible if it satisfies

d

a sensible metric then from (i) for every

V

for all

x

in

V,

and

is the zero element in $

Note that the given

||x|| d

+

||y||

||x||

=

for all

||x-y||

precisely

V; x, y

is recoverable from d(x,y)

=0

in

||•||

V. by the formula

.

45

A function

norm for

||*||

V,

and

linear space.

: V -> R

(1),

(2)

and

(3)

is called a

together with a particular norm is called a normed

V

Any normed linear space is automatically a metric space,

with metric defined by that this

satisfying

d

d(x,y)

does obey rules

=

||x-y||

(1),

(2),

:

it is straightforward to check

(3)

of the definition of a metric.

EXAMPLES of normed linear spaces

1. R

1 1x1

with

for n-vectors.

l

Li=l

1

1

.i 1

x

|2

\ ,

i.e.

the usual

'length'

function

y

This gives the euclidean metric and so it is called the

euclidean norm. 2.

f n

_

In particular, when

(a)

Rn

with

I |

(b)

Rn

with

11*11

X

| |

n = 1

Rn

||x||

=

|x|

.

= max{|x^ | = 11

+

1

In fact it can be proved that any norm on same topology on

we have

Rn

will give rise to the

as does the euclidean norm.

Of course the metric

will in general be different. 3.

The space of bounded functions f||

This

f

:

R,

x e R}

= sup{|f(x)

with

.

is a linear subspace of the space of all functions

(Example 3,

4. The space of bounded maps linear space and a number

R

R

§2.2) with pointwise addition and scalar multiplication,

the metric is that of Example 4,

K

equivalently,

A

f

§1.2. : A -> W,

such that

f (x)

f(A)cz B^.(0).)

'W

where

W

is a normed

(Bounded means there exists

is less than

K

for all

Again the obvious norm is

= sup{| |f(x) 'W

and

More generally:

is any set whatsoever.

f| |

46

R

x e A}

.

x

in

A;

5. The space of continuous | | f| |

- sup{| f (x) |

functions

f

:

[o,l]

-* R

with

Note that this is a linear space with

a < x $ b}.

pointwise addition and scalar multiplication as usual, sense in view of Theorem A of

§1.6.

This space is a linear subspace of

the special case of Example 4 above where norm and metric from this

differentiable at each point in

(0,1)

[jo,lJ ,

and it inherits the

f

:

[o,ll

-* R

which are

and whose derivative

Here we could take

= sup{ | f (x)

or,

A =

larger space.

6. The space of continuous functions

is a bounded function.

and the norm makes

f'

fI |

:

(0,1) -> R

where

a $ x $ b}

if we wished to measure derivatives as well as we would take where

= sup{|f (x)

a £ x < b} + sup{|f'(x)

a $ x $ b}

This is a linear subspace of the space in Example 5. naturally inherits is to work with is of Example 5,

'0

||•|

,

The norm which it

but the norm which it may be most useful is that

The metric corresponding to

1

•

§1.2.

7. The space of systems of differential equations

^1 = fi^xl»

x2') (F)

~

f2^xi’ x2^

defined on the closure that

f^,

f2

U

of a bounded open subset

U

of

R

and such

are continuous, where the linear structure is defined by

pointwise addition and scalar multiplication of maps f =

(f

f^)

|F||

:

R

2

-> R

2

,

and the norm is given by

= sup{||(f1(x),

f2(x))

x =

(x^,

x2)

e U}

47

Theorem C of that

||*||y

§1.6 shows that makes

U

sense.

is compact,

and then Theorem A'

It is routine to verify that

guarantees

||*||y

is a

norm. The corresponding metric is like that of Example 6,

§1.2,

although

2 there we supposed

f^,

f^

to be bounded on the whole of

R

Now we have essentially everything we need to pursue the study of differentiability.

However,

it is worth going a little further at this

stage to provide some machinery that will be useful later on.

Linearity and continuity Every linear map

L

The proof is easy:

:

let

and observe that if L(x)

Rn -* Rm

(with the euclidean norms)

e2>

(e^,

x

and

- L(y)

y

•••>

e^}

are in

Rn

be the standard basis for

by linearity

r n

l

Rn

then

= L(x-y)

= L

is continuous.

-v (xi - yi>ei

i=l n =

£ i=l

(x.

- y.)L(e.)

by linearity.

n Hence

| L (x)

- L (y) | 1

s

l

l*i - yi1

llL(ei)l |

$

||x-y||M

where

i=i n M =

l

l|L(ei)||

.

Therefore

||L(x)

- L(y)| |

less than

6 = e M ^

is less than

e

i=l whenever

1 1 x-y 11

is

.

In this proof we have used strongly the fact that the domain

Rn

is

finite-dimensional, but note that the proof would have worked with the codomain replaced by any normed linear space

48

F.

In general a linear map between two normed linear spaces need not be continuous.

Here is an example with, necessarily,

an infinite¬

dimensional domain.

EXAMPLE Let

V

be the space of all infinite sequences

— (^-^»

> a3 » . . . )

with addition and scalar multiplication defined term—wise,

the linear subspace consisting of those with

Define a norm for

E

Y lj, n=l

1

a

and let

E

be

finite.

n1

by oo

and define a linear map L(a^ »

Then

L

a2*

L a3♦

is clearly linear,

let

: E

E

•••)

—

by (a2>

Then

| |a^n^I I

= “

(n-1)!

but

• • •)

•

6

is there always exists some

less

6

but

||L(a^n^)||

= 1

in the

| |L(a^n^)| |

how small than

a^ >

but is not continuous at

denote the sequence with

elsewhere.

a3>

a^n^

0. nth

place and zero

,

and so no matter

= 1 in

To see this

E

with

| |a^n^ | |

.

The basic facts about continuity of linear maps are summarized in the following theorem.

The details of the proof are straightforward,

be found in any book

(such as Simmons

[ll8])

and can

on functional analysis.

THEOREM

Let

L

:

E -> F

be a linear map between normed linear spaces.

Then the

following statements are mutually equivalent:

49

(1)

L

is continuous.

(2)

L

is continuous at the origin in

(3)

There exists a constant

K

such that

is less than

K

for every

||l(x)||F

with

E

Statement the

(3),

If

L

K

.

S

in

E

F

F.

K

which satisfy

(3).

radius of the smallest sphere centred at the origin in

E

distorts

is a continuous linear map then there is a greatest

lower bound for all the constants

L(S ),

L

it nevertheless maps it inside some sphere

and centre the origin in

: E -*

in

x

can be described by saying that however much

'unit sphere'

of radius

1IxlI ii E = 1

E.

This F

is the

and containing

in other words sup{I IL(x)j |

This number is denoted by

| jL| |,

for it can be shown without much

trouble that it does define a norm on the linear subspace of

Lin(E,F)

consisting of continuous linear maps. Note that for such a map Lx

for every I |LI I

x

in

by writing

E.

L

we have the inequality x

'F ^

This follows immediately from the definition of

I Ixl I

E

= a

and applying

| |L(a 1x)| |F ^

since

II a

Ell

linearity of

E

L

The notion of

= 1 ,

L

to

(1)

we have

| |L | |

but the left hand side is

and property

a ''"x :

a ^||l(x)|| 1 1

1 1 ]7

by the

J

of the norm.

the normed linear space of continuous

linear maps

E -v F

with particular properties of the norm such as the inequality above, will

50

,

be important in §2.6 when we consider higher orders of differentiation. Quite apart from these uses,

however,

it constitutes one of the basic

pieces of equipment in functional analysis.

Completeness Recall

that

in a metric space

converges to

c

in

given any m If

S

S

we say a sequence

d(x

n

,

c)

< e

for all d(x

n

with

,

c)

|x

n > m.

=

n'

this is just

...

there exists a positive integer

is a normed linear space we have

S = R

x^,

when:

e > 0

such that

S

x i

- c n

1 1

,

and if

n

When we are given a sequence how can we tell whether or not it converges to some d(x^,

c?

c)

If we have a plausible guess for

c

then we can look at

and try to check the definition, but what methods are available

if we are not sure even whether is a very useful lemma

c

exists?

S = R

there

(often known as the General Pr'inciple of Convergence)

stating that a sequence of numbers something)

In the case

x^,

x^,

...

converges

(to

if and only if

given any m

e > 0

such that

there exists a positive integer

x

< e

- x nl

for all

n^, ^ > m.

n2

Unfortunately the generalization of this to arbitrary metric spaces with d(x

, x nl

example,

)

replacing

x

n2 S

take,

for

n2

to be an open interval or to be the set of rational numbers

with the metric induced from S

turns out to be false:

- x nl

R.

will converge to something in

In these cases a sequence as above in R

which may not belong to

S.

51

Metric spaces

in which this General Principle of Convergence does hold

are sufficiently important and common for it to be worthwhile identifying they are called complete metric spaces.

them with a particular name:

A normed linear space which is also complete when viewed as a metric space is known as a Banach space

(S.

Banach,

1892-1945).

The examples given above of non-complete metric spaces are, not

(real)

linear spaces.

a finite-dimensional

of course,

In fact it would be impossible to construct

linear example in view of the result that every

finite-dimensional normed linear space is automatically a Banach space. Completeness of a normed linear space is a technical condition needed for the correct statement of some general results later on,

but in finite

dimensions there will be no need to worry about it explicitly.

Hilbert spaces Apart from its linear and norm structures,

Rn

has

interesting geometrical

properties that relate to the inner product

= *^1 + x2y2 + of pairs of elements for vectors in

R

3

.

+ xnyn

familiar as the dot product or scalar product

x, y

We have

=

i i 2 ||x||

,

and in

R

3

it can be

shown that = where

0

||x||

||y||

is the angle between the vectors

this as the definition of

0,

cos 0 x, y.

In

Rn

we can take

and mimic much of the geometry of

RJ

.

In functional analysis there are many circumstances under which a Banach space comes provided with an inner product analogous Rn

.

Technically,

means a map

52

to that in

an inner product on a real normed linear space

: E x E -> R

E

which is linear in each factor separately,

is

symmetric

for every

( = ),

x,

and is such that

vanishing only when

x = 0.

are some amendments to these conditions.) it

(In the complex case there Using such an inner product

is possible to derive much stronger results about the structure of a

Banach space than would be possible without it. an inner product is called a Hilbert space (D. Sometimes

sense.

R) .

Hilbert,

1862-1943). although

is still not too large in a topological

space is called separable if it contains a countable

A topological

dense subset

A Banach space which has

it is necessary to ensure that a Hilbert space,

possibly infinite-dimensional,

in

is non-negative

(as with the countable set

Q

of rational numbers contained

It is particularly useful to deal with separable Hilbert spaces,

since it can be proved that they have in a certain precise sense at most countably infinite dimensions,and many finite-dimensional techniques can be extended to this case by replacing n-tuples

(x^,

..., x^)

by

infinite sequences.

2.4

DIFFERENTIATION

Let

E

||.||

and

F

| | . | |f,

linear) map,

(continued)

be two normed linear spaces with respective norms suppose

and let

x

f

: E -> F

is a given continuous

be a particular point of

E.

(not necessarily

By analogy with

§2.1 we make the following definition:

DEFINITION

The map

f

is differentiable at

which approximates

f

at

f(x+h)

x

x

if there is a linear map

in the sense that for all

- f(x)

= L(h)

+

h

in

L

: E E

F

we have

||h||E h(h)

53

where

is an element of

n(h)

F

with

| | n (h) | |

as

-> 0

In general

L

of course depends on

we should write something like the role of

|h|

in §2.1

L

while

longer a number but an element

Here

plays the role of

(thought of as L =

'small')

exists

It is a linear map from

Df(x).

L.

of

E

to

f

Df(x)

instead of

Df(x)(h).

is a linear map from

Remember that E

to

F,

h

and so

e

which is no

F.

at

plays

It is not E, F, x

f

and

and denote

and not a number.

F,

To avoid proliferation of brackets we will often write Df(x)h

||h||

(for given

We call it the derivative of

then it is unique.

it by

ij

and so to be formally correct

instead of n(h)

difficult to show that if such an x)

x,

0 •

||h||

r

Df(x).h

or

is an element of Df(x)h

E,

is an element of

F. Using this notation we can re-write the defining expression for the derivative as f(x+h)

where

| |h| |

means

- f(x)

= Df(x)h +

j |h| |^,

and

||h||n(h)

p(h) -*■ 0

(in

The pictorial interpretation is much as before,

(*)

F)

as

cartesian product image of

f.

Df(x) f(x)

See Figure 9.

54

E x F)

(which will be some kind of

is a linear subspace of

it is

E).

Instead of 'surface'

in the

it can often be more useful to visualize the

This consists of some distorted version of

The image of the vector

f

(in

although the spaces

now may not be 1-dimensional or even finite-dimensional. thinking of the graph of

h -*■ 0

'tangent'

F,

to the image of

E

inside

F.

and when translated by f

at the point

f(x).

image of f

Frgure 9 Equivalently,

the map h

f (x)

+ Df (x)h

is a first order approximation to the map

h i-»- f(x+h)

Interpretation of the derivatives in familiar cases We will now look at the meaning of the above rather abstract definition in the case of maps between euclidean spaces, where differentiation is a more familiar concept.

Case 1. Here

f

E =

R,

F =

Rm . R -> Rm

is a continuous map

,

which we call a path

in

Rm .

(From some points of view it would be more sensible to give that name to the image of

f

parametization by At each point map

R -> Rm

.

R™

in

t

R. in

but then we would have lost track of its

We shall continue to call the map a path.)

R

the derivative

Now linear maps

very easy to understand, so by linearity of L(s)

,

L

L

all you have to do is

L(s)

to find

R

from

since for any

we have

Df(t),

s

is a linear

to any linear space in

= sL(l). L(l)

if it exists,

R

we have

s

= s.l

In other words,

as a element of

F

F,

are and

to find and

55

multiply it by the scalar L(l)

=0

in

the zero map

s.

Thus there are two possibilities:

F,

in which case

R ■+

F)

or

L(l)

L(s) / 0

= 0

for all

R -* Rm

L = Df(t),

then its

translated to

we see that if

L(l)

Df(t)

image is a straight line which,

f(t),

(i.e.

in which case the image of

straight line through the origin and the point the case when

R

s e

is tangent at

f(t)

in

F.

either L

L

is is

the

Returning to

is not the zero map after the origin is

to the image of

the path

f.

See Figure 10.

Figure 10 The association of

L(l)

R

between linear maps

with

L

sets up a 1-1 correspondence

and elements of

-> F

F,

and it is a simple matter

to check that this is a linear isomorphism between the space of all linear

R

maps F

-> F

itself.

(pointwise addition and scalar multiplication) For this reason whenever we have a map

particular when but

Df(t)(l)

f'(t).

F = or

Rn)

which is an element of

By linearity we have for all Df(t)s = Df(t)(s.l)

f(t+h) where

56

n(h) -> 0

as

(*)

:

R

-> F

(in

it is often convenient to work not with

Df(t)l,

and so the formula

f

and the space

s

in

h -* 0.

and denoted by

R

= sDf(t)l = sf'(t)

defining the derivative of

- f(t)

F

Df(t)

= hf’(t)

+

Finally,

f

:

R

F

becomes

||h||n(h) returning to the specific case

when

F

R

we have

formula shows us that (f

f(t)

-

(f^(t),f2(t),...,fR(t))

f'(t)

'(t) , f 2'(t) ,. . . ,f(t) )

is simply the n-vector where the

the real-valued functions

f^.

Of course.

f(t)

f^1

are the usual derivatives of

This fits neatly into Figure 10,

illustrating our familiar notion of vector based at

and the above

(f,'(t),f„'(t),...,f '(t)) l z n

and tangent to the

(image of

the)

as a

path there.

Figure 10 serves equally well as a pictorial representation of

the case for general

F,

not necessarily

Rm.

To round off this description we look at the case

n = 1,

bringing

us back to the beginning of the discussion of differentiation (§2.1). We have two concepts of derivative:

the old familiar

and the new

R -* R

Df(t)

as a linear map

.

f'(t)

as a number,

What is their relationship?

From the above we see that it is merely that f'(t)

= Df(t)1

the right hand side being a number

E = R

Case 2. Here

f

,

,

(element of

since

= x,

and

Df(x)

(if

n

real variables

it exists)

is a linear map

To analyze this linear map we will take the standard basis for

Rn

(see

§2.2).

Then any

h = h^ + h2e2 +

(i.e.

h =

(h

n = 1.

F = R .

is a real-valued function of

(x15x2,...,xn)

R)

,h2,...,hn))

h

in . . .

Rn

Rn

R

{e^,e2,...,en>

can be written uniquely as

+ hnen

and so by linearity we have

Df(x)h = h1Df(x)e1 + h2Df(x)e2 +

= h-^a-^ + l>2a2 +

•••

+ ^nan

...

»

+ hnDf(x)en

saY'

57

By taking

h

to be of the form

place and zeros elsewhere) formula

for each

(*), we find that the 3f

partial derivatives

(0,0,...,1m,...,0)

cm

i

in turn,

tm

in the

ith

and substituting in the

are nothing other than the familiar

at the point

3x.

(i.e.

x.

1

If

v

is any element of

(partial)

derivative of

R

f

then the number

at

x

Df(x)v

in the direction of 3f

terminology the partial derivative

(x)

3x.

is called the v.

Thus

is the derivative of

in this f

at

i

x

in the direction of

linear space

E

e^.

If

R

then the notion of

is replaced by an arbitrary normed 'standard'

3f

partial derivatives

3x. l

will not exist, but the derivative of v

is an element of

Case 3.

E = Rn,

In this case

and

Rm

in the direction of

(where

will still make sense.

is a linear map

^el»e2’'*"’em^

Rn

R™

,

and

so if we choose bases

(not necessarily the standard bases)

respectively we can represent

Specifically,

v

F = Rm .

Df(x)

{el,e2,•••,en}, Rn

E)

f

Df(x)

for

by a matrix.

if h = £, e-.

11

+ £„e0 +...+£ e 2 2 n n

II

+ A„e„ + 2 2

then Df(x)h =

where the

A's

are related to the

...

£'s

+ A e mm by

n Ai =

E “ii j=l J

for a collection of and

58

n

columns.

mn

£i

»

scalars

1 $ i $ m

{om^}

forming a matrix with

m

rows

If we now do take the bases to be the standard bases,

l.

so that the

1 and

are the usual coordinates, f(x)

=

then writing

(f1(x),f2(x),...,fm(x))

and putting together the discussions of cases 1 and 2 we find that

3f. ij

for each

i,

j.

standard bases

The matrix representing

9x^

8f2 9x^

m 9x,

is

Df(x)

with respect to the

is thus

K

This

(x)

9x.

9f (x) 9x2

3f2

(x)

9x2

3f (x)

m

3x,

(x)

9x

(x) n

3f (x)

(x)

8x

3f (x)

m

9x

(x)

the so-called Jacobian matrix (with respect to standard bases)

condensed to the notation

3(f1> f2» ‘ ' ‘ ,fm^ 9 (x-^ ,x2 , • • • 9x^)

Observe that we could equally well construct an analogous Jacobian matrix for other choices of bases. When

m = n

the matrix is a square matrix,

This number is often denoted by understood:

Jf(x)

or just

and so has a determinant. J

when

it will be of considerable interest to us

f

later,

and

x

are

in §2.8.

59

Remarks 1.

The Jacobian matrix depends on a choice of bases, whereas

derivative

Df(x)

does not.

the

The derivative was defined entirely

without reference to coordinates. 2.

Although the differentiability of

f

at

the partial derivatives of the components fact that the converse does not hold. partial derivatives

x f^

This

implies the existence of of

f,

it is a standard

is essentially because the

tell us approximately how the function behaves in

particular directions, whereas the derivative behaviour in a whole neighbourhood.

Df(x)

deals with the

It can be proved,

however,

that if

the partial derivatives exist and are themselves continuous functions some neighbourhood of

x

then

f

n

is differentiable at

x.

in

We shall not

be concerned with technicalities of this kind in what follows. 3. Digression into the complex field.

All

the above ideas work just as

well for maps between complex linear spaces.

In particular,

if

Cn

denotes the linear space of n-tuples of complex numbers with co-ordinate wise addition and multiplication by scalars

(complex numbers)

and norm

given by

r n

\

l I z.

2

1 2

1

'•i=l

then the derivative of a differentiable map

f

complex-linear map

Cn

R2n

by,

Df(z)

for example,

:

Cn -* C™ .

Now

identifying

(x1,y1,x2,y2.xn,yn)

where

:

= x.

+ iy.

J

J

J

at

C

for each

j

= l,2,...,n

we can multiply by complex scalars while in

multiply only by real scalars. real

60

linear map

R2n

R"m

A complex-linear map

(since real

will be a

with

n although in

z

can be thought of as

...,z )

z.

Cn -> C™

Cn -* C™

o RZn

,

we can

becomes a

scalars are special cases of

complex scalars)

but the converse is not true.

conditions for a real-differentiable map complex-differentiable map 9u.

9y. J

where we write

k

Cn -* Cm

,v. k

*

9y. J

:

z

in

Cn

R^n

the

to represent a

are that

1 $ i $ m

1

9x. J

u

:

9v.

9v.

_1

each

f

f

At

at

9x. J

z,

1 J? j $ n

f (X]_ ^ ,x2 ,y2, . . . ,xn,yn)

being a function of the

=

x.

(U]_ ^ ,u2 ,v2 , . . . ,um,vm) and

l

y..

,

These are the

J3

classical Cauehy-Riemann equations,

So far we have been discussing maps F

E.

x

it is not necessary that

All we need is

sufficiently small of

whose domain

E

and codomain

are both normed linear spaces, but it is clear that for differentiability

to be defined at of

f

h,

that

f(x+h)

i.e.

U

such that

that although the formula

x + h

lies

in

U,

the map

can be thought of as the whole of f

B^(x)

•

Hence if the domain

in a normed linear space

sense to talk about differentiability of Note, however,

be defined on the whole

should be defined for all

on some 6-ball

is given as an open set

f

f

E.

: U

f

at any point

(*)

E x

it makes in

only has meaning if

Df(x)

U. h

is

is linear and so its domain

Thus associated to

F

we have Df (x) for every If it is

x

in

U

the case that

then we simply say

differentiable on Df (x)

at which

: E -*■ F,

f U.

: U

f

: E -* F f

(linear)

is differentiable.

is differentiable at F

is differentiable,

For each

x

in

U

x

for every

or that

f

x

in

U

is

we have a linear map

or in other words we have a map

61

Df

: U -> {linear maps

E -* F} = Lin(E,F)

At this point we may wish to ask whether the derivative of varies continuously with continuous. Lin(E,F), of

x,

i.e. whether the map

at

x

U -*■ Lin(E,F)

:

is

To make sense of this we would have to have a topology on or at least on some subset of

Df.

Df

f

Lin(E,F)

containing the image

Now it so happens that if the continuous map

differentiable at

x

then

Df (x)

is a continuous linear map

(i.e.

This means that the image of subspace of

Lin(E,F)

Df

f

:

U

is not merely a linear map continuous on :

U -> Lin(E,F)

consisting of continuous

in §2.3 this space, which we will now denote by

E, is

F

is

E -*■ F

for each fixed

but x).

contained in the

linear maps. L(E,F),

As we saw

has a natural

structure as a normed linear space and hence a metric space. Therefore if we regard

Df

not just as a map

Df

:

U -> Lin(E,F)

Df

: U -> L(E,F)

but as a map

we are able to discuss whether or not it is continuous.

DEFINITION

If

Df

is continuous we say that

is of class

This

on

f

: U -> F

-or simply that

U

is of class is

f

C^

c\

or

f

.

somewhat abstract formulation may seem far removed from familiar

ideas of continuity of derivative in euclidean space, but the gap is easily bridged. between map

62

Lin(Rn,Rm)

Rn -* Rm

a norm on

To begin with,

is

and

L(Rn,Rm)

continuous.

L(Rn,Rm).

recall that since,

there is no distinction

as we have seen,

every linear

There are various reasonable ways to put

First of all we can take the norm defined above in

the general context, namely |L| |

= {sup| |L(x)

M

= 1}

where the norms on the right hand side refer to the usual euclidean norms.

Rn, Rm

Secondly, we could choose norms other than euclidean norms for and make the analogous definition of bases for

Rn

A = (a„)

and, regarding

and

Rm,

the euclidean norm of

||l||.

then represent A

A,

L

Thirdly, we could choose by an

as an element of

m x n

R™1,

take

matrix ||l||

to be

i.e. f m

n

l

l

a. . ij

*4=1 j=l

Then there are other possible variants such as 1 lLl 1

= max

1

i> j and so on.

a. . 1 ij 1

Now the important fact to use here, which sweeps away all

this confusion, is that it makes no difference which norm we choose on

L(Rn,Rm).

More precisely,

since

L(Rn,Rm)

invoke a non-trivial theorem (Simmons

is finite-dimensional we can

[ll8j) which says that any two norms

on a finite-dimensional linear space will induce the same topology, although of course the induced metrics will in general be different. Consequently, for studying the continuity (a purely topological property) of measuring

Df

: U •*

||Df(x)||.

L(Rn,Rm)

we can choose whichever way we like of

A convenient way is to take standard bases and let 3f •

Df (x)

max i» J

1

3x. J

(x)

It is then simple to see that continuity of

of the partial derivatives

3f. ^ - .

Therefore

Df

is the same as continuity

f : U -*

R

(U

an open

Xj

63

3f. subset of

R )

is

precisely when all the

C

are continuous

Xj functions on

U.

This freedom in the choice of norm is another illustration of the advantages of working with topologies rather than particular metric structures.

2.5

PROPERTIES AND USES OF THE DERIVATIVE

Unless with

stated otherwise, U

E

and

an open subset of

F

will denote normed linear spaces

E.

The following properties of the derivative can be verified directly from the definition:

Linear combinations.

(i)

is the map

af + Eg

If

: U ->- F

for any constants

D(af + Bg) as maps

f,g

U -*■ L(E,F).

are differentiable then so

a,3,

and

= aDf + BDg

Constant maps themselves have derivative zero,

of course.

(ii)

Linear maps.

linear map

L

If

: E -> F

f

is the restriction to

then

Df(x)

= L

already its own linear approximation. (Note that we do not say In the special case and

(iii)

Df(x)

= Z

Dt(x)

E = F = R

Bilinear maps.

If

B

x

in

U,

i.e.

f

is

C

L

is

.

which would not make sense.)

we have

E = E^ x E2

of a continuous

In particular

regarded as the linear map

of a continuous bilinear map 64

= L(x),

for every

U

f(x) R

and

: E^ x E^ + F

= Zx R

f

:

x h- Zx

.

is the restriction to (i.e.

l,

for some number

B

U

is linear in each

factor separately)

then

Df(x)h = B(Xl,h2)

where we write in

E2.

x =

(x^x^,

In particular

taking

x

f

+ B(h1,x2)

h = is

(h^t^) c\

with

x^l^

since the map

in

Df

E1

and

x2,h9

: U -* L(E,F)

to the linear map (h

B(xrh2)

+ B(h1,x2))

is continuous. As an example,

consider

E^ = E2 = F = R

(ordinary multiplication).

and

f(x

,x2)

= X]x2

Then

Df(x)h = x1h2 + h1x2

=

(x2,x1)

which is another way of saying that

Cartesian products.

(iv)

If

differentiable then so is (f^

x f 2) (x-j-,x2)

means

(f^Cx^),

of

f f

:

U

F

lies in

f?

:

U2 -> F2

x U2 -* F^ x F2

f2(x2)),

(Df1(x1)h1,

(Important)

normed linear spaces, Let

=

and

are

where

and we have

Df2(x2)h2)

.

derivatives operate co-ordinate-wise.

Compositions.

(v)

-* Fx

f = f^ x

Df(x1,x2)(h1,h2)

In other words,

3f 3f - = x„, - = x ox^ l 3x2 1

and V

and g

:

U,V V

G

Suppose

E,F

are open sets in

and E,F

be continuous maps,

so that the composition

g*f

G

are three

respectively.

and suppose the image

: U -* G

exists:

see

Figure 11.

65

Figure 11 If g*f

f

is differentiable at is differentiable at

x x

D(g* f)(x)

and

g

is differentiable at

f(x)

then

and = Dg(f(x))*Df(x)

Observe that the right hand side is the composition of the two continuous linear maps

: E -*■ F

Df (x)

and

continuous linear map from

Dg(f(x))

E

to

:

F -> G

and is

itself a

G.

This formula for the derivative of

g*f

is known as

the Chain Rule.

It can be expressed by saying that the derivative of a composition is the

composition of the derivatives, but note that care must be taken to specify at which points the derivatives are calculated. In euclidean space the derivatives Jacobian matrices, derivatives. y = f(x),

and then we obtain the familiar chain rule for partial

Explicitly,

z = g(y),

if

E =

Rn,

F

= Rm,

G =

R^3

and we write

then the chain rule states that

9(Zi,z2,..•,zp)

= 3(z1,z2,...,zp)

3(x15x2,...,xn)

3(y1,y2,..•,ym)

or equivalently 3z.

m

3z.

-a= y _i 3x. . 3y, j

66

can be represented by corresponding

k=l

Jk

^k 3x. J

3(y1»y2»---»ym) ‘

3(x1,x2,...,xn)

which is

the form in which the rule is most commonly expressed.

This form is the one which is most useful for specific calculations, but its disadvantages are

Rn

co-ordinates in

(1)

it is

etc.,

(3)

cumbersome, it gives

(2)

it depends on choices of

(as written)

no information about

where to evaluate the partial derivatives.

EXAMPLES 1. m

Ordinary multiplication of real numbers is a bilinear map :

R

R,

x R ->■

and the Chain Rule allows us to use this to derive a rule

for the derivative of the pointwise product functions

f

m(f(x),g(y))

:

R

E

and

g

we can write

D(fg)(x,y)

:

R.

F -*

: E x F -*

Expressing

fg = m*(f x g)

R

f(x)g(y)

of two as

and so by the Chain Rule

= Dm(f(x),g(y))

•

D(f x g)(x,y)

= Dm(f(x),g(y))

*

(Df(x)

by the rule for cartesian products. E x F

fg

x Dg(y))

Applying both sides

to

(h,k)

in

we obtain D(fg)(x,y).(h,k)

= Dm(f(x),g(y))•

(Df(x)h,Dg(y)k)

= m(f(x),Dg(y)k)

+ m(Df(x)h,g(y))

by the rule for bilinear maps. f(x)Dg(y)k + g(y)Df(x)h

The right hand side is

simply

(multiplication is commutative),

and so we can

write the conclusion as D(fg)(x,y)

= f(x)Dg(y)

where here the right hand side is two linear maps case when

Df(x)

E = F =

R

:

E -

R

+ g(y)Df(x)

a pointwise linear combination of the

and

Dg(y)

:

this of course reduces

F

R.

In the particular

to nothing more than the

usual product rule (fg)'

= fg'

+ gf'

67

2.

If

f(x)

B

: E x E ->

linear map

f d

is a bilinear function let

f

:

E ->

R

be defined by

Such a function is often called a quadvat'ic fovm.

= B(x,x).

We can write

R

as the composition

:

E -* E x E

f = B*d

taking x to

Df(x)h = DB(d(x))

•

(x,

where

x) .

d

Since

is

the

Dd(x)

'diagonal'

= d

we have

Dd(x)h

= DB(x,x).(h,h) = B(x,h) If

B

+ B(h,x)

is symmetric in the two factors this becomes Df(x)h = 2B(x,h)

or

Df(x)

and

= 2B(x,*)-

B =

A particularly common example is

scalar product

,

the case

E = Rn

i.e. n

B(x,y)

= x.y =

\

x.y.

:

i=l 2 here

f(x)

=

||x||

can represent by the f(x)

and

Df(x)

1 x n

is the quadratic form T

(where

is

the linear map

h h- 2x.h

matrix or row-vector T x Ax

then

x

This we

More generally,

. is the row-vector

Df(x)

denotes transpose of the vector

2x.

.

and of the matrix

if

T T x (A+A ) A) .

3. A superficially more complicated but in fact equally straightforward class of examples is illustrated by the following: m x n

matrix and an n-vector,

respectively,

with respect to some real parameter What is the formula for Writing

f = B • B

we have

68

:

f'(t),

(A x b)

where

L(Rn,Rm)

x

t,

Let

Rn + Rm

and let

f(t)

(A,b) h- Ab

,

= A(t)b(t)

Df(t).l,

is the bilinear map :

be an

each varying differentiably

or, more precisely B

A,b

in

R™ .

in

Rm

?

f'(t)

= Df(t).1 = DB(A(t),b(t))

•

= B(A(t),b'(t)) = A(t)b'(t)

which makes in

L(R

sense because

,R ).

b'(t)

If

f

inverse Df(x)

g

:

:

■+

U

.

E

+ B(A'(t),b(t))

+ A'(t)b(t)

= Db(t).1

in

Rn

and

A’(t)

= DA(t).1

There are many similar examples that could be constructed

using products of matrices,

(vi)

(A'(t),b'(t))

V

V ->■ U F

transposes and so on.

is a differentiable map which has a differentiable then at each point

is a linear

x

isomorphism.

g*f = id

:

in

U

the derivative

This is easy to prove:

we have

U -* U

and so the Chain Rule gives Dg(f(x))

•

Df(x)

= D(id)(x) = id

since

id

:

U -> U

is

(the restriction of) f • g = id

:

V

Df (x) Df(g(f(x)))

inverse, Again,

and

•

Dg (f (x))

= Df(x).

so in particular

E a linear map.

Similarly

V

and so by the Chain Rule at the point

since

: E

f(x)

= id

:

Therefore Df(x)

in

V

F -> F Df(x)

has

Dg(f(x))

as its

is a linear isomorphism.

this can be casually expressed by saying

the derivative of the

inverse is the inverse of the derivative. A differentiable map with differentiable inverse is called a

diffeomorphism.

A diffeomorphism defined on some open neighbourhood

of a given point

x

is called a

local diffeomorphism

at

x,

U

although this

term is often reserved to apply to local diffeomorphisms whose domain and codomain are both open neighbourhoods of

x

in

E

and which keep the

69

point

x

itself fixed.

simply as invertible

(vii)

Local diffeomorphisms at

(non-linear)

roughly,

can be regarded

changes of local co-ordinates near

A much-used result in elementary calculus

This states,

x

is

Mean Value Theorem.

the

that if a line segment is drawn joining two points

on the graph of a differentiable function then there is

somewhere on the

graph in between these points where the slope of the curve is the slope of the line segment. [x,

x+h]

x.

More accurately,

lies in the domain of

the same as

if the interval

the differentiable function

f

(although

strictly speaking we do not need differentiability at the end-points: continuity there suffices) x < E, < x+h

then there exists some point

£

with

such that f'(0

=

(f(x+h)

- f(x))/h

.

Rewriting this as f(x+h)

= f(x)

+ f' (£)h

gives another interpretation of expressed exactly as map is

f(x)

the derivative of

the theorem,

namely that

plus a linear map applied to f

at

E,,

not that at

definition, may give only an approximation to interpretation, n

f(x+h)

rather than the

'graph'

x

which,

f(x+h).

version,

h.

can be

The linear from the

This

latter

generalizes directly to

dimensions or to any normed linear space:

THEOREM (Mean Value Theorem)

Let U, U.

f

:

U -+ R

be a differentiable function.

and suppose the whole line-segment joining Then there is some point f(x+h)

70

= f(x)

y

Let x

x

to

and x+h

x+h

also lies in

on the line-segment such that + Df(y)h

lie in

The proof is easily obtained from the one-dimensional version above. Given

x

and

h

in

E,

(t)

let

R -+ R

be the function defined by

= f (x + th)

i.e.

f

evaluated at the point

from

x

to

x+h.

:

Then

a

proportion

t

along the line segment

is differentiable, being the composition of

the differentiable functions

R -+ E

:

t

x + th

and

f

:

E -> R,

and

the Chain Rule gives '(t)

= Df (x + th)h

By the standard Mean Value Theorem the left hand side equals for

t

equal to some (f)(1)

and so taking

£

between

- F

is of class

is

to say that the

map Df is continuous, where |L| |

: U ->■ L(E,F)

L(E,F)

has the topology which comes from the norm = 1}

x

= { sup | | L (x)

(Note:

Do not confuse the possible continuity of

(if

is continuous)

f

whether is,

Df

then

continuity of

Df(x)

D(Df)(x)

.

2

2

We can next ask

.

U.

If it

Df

f

at

x,

abbreviated to

implied that its derivative was a

so the fact that

.

If

in

.....

f

Df

is continuous

.2

is a continuous linear map,

L(E,L(E,F)).

x

.

Just as the continuity of

continuous linear map, D f(x)

: E -> F.)

will be a linear map

is called the second derivative of

D f(x).

with the necessary

itself is differentiable at a given point

E + L(E,F) This

Df

i.e.

D f(x)

implies

that

belongs to

is differentiable on the whole of

U

we thus have

a map D2f

: U -> L(E,L(E,F))

If this is continuous, which is we say that

f

is of

class C

2

.

the same thing as .

saying that

Df

is

C

Although the right hand side looks

complicated it is still a normed linear space,

and we can therefore

. . # . continue m this way to define

If

C

3

,

C

4

etc.

f

is of class

C

t

00

for all

r = 1,

2,

3,

Clearly,

. if

. is

C

f

... r

then we say that

then

•

D

r~l

f

*

exists •

•

f

C°

72

C

1C

•

•

implies

to mean continuous.

C

s

for all

s < r.

C

and is continuous,

would not have considered differentiating it: and hence

is of class

thus

C

1

or we

implies

C

Conventionally we write

1

,

,

Sometimes

literature to mean 00

C

smooth

the words C

.

or just

differentiable

are used in the

smooth

Later we shall tend to use

to mean either

27

or

C

for some

r

understood from the context.

Discussion of derivatives of high order rapidly becomes very uncomfortable because we have to deal explicitly with spaces of type

L(e,L(e,L(e,...,L(e,L(e,f) ))...)

.

Fortunately we can avoid the problem by making use of some linear isomorphisms which conveniently exist between these spaces and a family of other types of space which are conceptually easier to grasp. spaces of

These are

multilinear maps.

We have already encountered examples of bilinear maps. An

n-multilinear map M :

where all

the

E^

is

E, x E„ x 12 and

each factor separately are held fixed).

a map

F

...

x E

n

F

, ’

are linear spaces, which is a linear map in

(i.e.

when all the components

For example,

the multiplication map

(x,,x„,...,x ) h- x.x„...x 1* 2 * n 1 2 n

is an n-multilinear map r

Rn = R x R x

As another example,

n x n

...

x R -* R.

times,

...

x Rn

each factor corresponding to one column vector.

determinant

Then the

map is an n-multilinear map Rn x Rn x

As

consider the space of all

matrices regarded as Rn x Rn x

n

in the other factors

...

x Rn

R

special cases we note that a linear map is a 1-multilinear map

and a bilinear map is a 2-multilinear map. n-multilinear map is in general multiplying something on the

not linear,

Observe that for

n >

1

an

since the effect of

left hand side by a scalar

a

is

to

73

multiply in

F

by

oin.

Consider now the space L (E^ , L(E^,F)). every E2

in

the map

Writing

An element E^

L(x^)

and

bilinear.

instead of

:

L

L(x^)

L(x^)

in

L(E2»F).

L(x^).X2

were

Now to every

L(x^).X2

of

X2

in

F.

we have constructed a map

linear this map

since

L(x^)

is continuous, where

product topology Conversely,

this space is a map associating to

E^ x Ej 4 F

Furthermore,

shown that

of

or more generally

associates an element

L(x^,X2)

L

L

an element

L

Since

L(E,L(E,F)),

L

and

is easily seen to be

L

are continuous

x E2

it can be

is equipped with the

(see page 15).

if we start with a bilinear map

L

:

E^

x E2 -*■ F

then we

can define a corresponding linear map L

by taking taking

L(x^)

X2

to

continuity of

to be,

turns out that L (E^,L(E2,F)). between

L

M2

or,

However, L(x^)

x^

E^,

if it is,

that

L

: E^ -> L(E2>F)

if we do assume that and

L

L

L L

F

L(x^)

will be

L

belongs

to

gives a natural bijection

M2(E^ x E2;F)

of continuous

and moreover it is routine to verify that when

is made into a linear space by pointwise addition and scalar

multiplication this bijection is a linear isomorphism.

74

E2

is continuous then it

are continuous and so

and the space F,

the linear map

there is no reason to expect that

Therefore we see that

E^x

in

Now since we have assumed nothing about

to begin with,

L (E^,L(E2,F))

bilinear maps

for each

L(x^,X2).

will be continuous, continuous.

: E^ -* Lin(E2 ,F)

In a similar way,

the space

L(E1,L(E2,L(E3,...,L(En_1,L(En,F)))...)

can be seen to be linearly isomorphic to the space M

n

(E, x E-> x 1 2

...

x E

n

,F)

of continuous n-multilinear maps with pointwise addition and scalar multiplication:

the correspondence is

L(x15x2,...,xn)

=

L L

where

(...((L(x1))(x2))...(xn_1))xn

.

This now gives us a much less clumsy notation for dealing with derivatives.

Instead of (...((Drf(x)(h1))(h2))...(hr_1))hr

we simply write

9

D f (x) (h.^ 5^2»• * •

thinking of When

Drf

E = Rn

U -*■ M^(E x E x

here as a map

and

F = R,

...

x E;F).

so that we are dealing with a real-valued j-

function of

n

variables,

the expression for

D f(x)(h,h,...,h)

becomes a. . ij

(x) •P

h.h....h 1 J P

where 3rf ii...p

3x.3x....3x i J P

and the summation is over all r-tuples integers

from

1

to

n.

When

n = 1

(with possible repeats) this reduces

of the

to

Drf(x)(h,h,...,h) = f(r)(x) hr where

f^

is the usual

rth

derivative of a function of one variable.

Formally, we see that the relationship between the abstract interpretation of

Drf(x)

f

(x)

as a number and

as a multilinear map is that

75

f(r)(x)

= Drf(x)(l,l,...,l)

.

In other words, we need only one coefficient to describe the effect of an r-multilinear map M = Drf(x)

M:RxRx...xR->-R

this coefficient is

We remark here that,

on

(h,h,...,h),

and when

f^r^(x).

as with linear maps,

every multilinear map is

automatically continuous when we work with finite-dimensional spaces. Therefore explicit references to the continuity of

D f(x)

multilinear map can be conveniently dispensed with when

as a

f

is a map

between euclidean spaces.

The particular case map

E x E

form on

Rn

F. .

r = 2.

When

Here

E = Rn

D f(x)

and

F = R

Given a choice of basis

Rn x Ru -> R

in

is thought of as a bilinear then

D2f(x)(*,*)

Rn

any such bilinear map

can be represented by a matrix

B(u,v)

,

Q =

(q^^),

is

a quadratic

where

= u Qv n u.q..v. i W J

i»j=l When

B = D f(x)

this matrix

Q

of second partial derivatives of

d2f (x) 9x^9x^

92f 3x 3xn n 1

This

f

at

92f

(x)

92f 9x 9x„ n 2

9x.. 9x 1 n

in

,n R ,

92f

(x)

is called the Hessian matrix of

x)

9x 9x n n

f

is

the matrix

x:

9x^9x2

(x)

choice of co-ordinates

76

92f

(now depending on

at

x.

(x)

(x)

It depends on the

whereas the original bilinear map

2 D f(x) It

:

E x E -> R

does not.

is well known that for a point

minimum for

f

3f gx

, v vxJ >

/__\ 9f vx/> gx 12

x

to be a local maxi-mum or

it is necessary that the partial derivatives 9f

, N (.x)

...»

should all vanish,

and then in order to

n

determine whether

x

is a maximum or is a minimum it is necessary to

look closely at the properties of the Hessian matrix at return to this

x.

We shall

in detail in §2.9.

It is a standard fact from elementary calculus that when f

:

Rn -* R

is

C2

we always have

2

9 f 9x.9x. i

i,j

2

/ \ (x)

9 f -w 9x. 9x.

=

1

- l,2,...,n,

J

which means

(x)

i

that the Hessian matrix is symmetric.

2 This is

is

the same as

symmetric,

saying that the bilinear map

u,v

:

Rn x Rn + R

i.e. D2f(x)(u,v)

for all

D r(x)

•

in

R

n

.

= D2f(x)(v,u)

More generally,

if

f

is

C

it

then each

rth

partial derivative is independent of the order of differentiation. The corresponding co-ordinate-free version of this is that the r-multilinear map ordering of the

D f (x) r

is symmetric,

components of

can be found in Dieudonnd

[32^,

i.e.

independent of the relative

ExEx...xE.

A proof of this fact

for example.

Note in passing that of the two examples of multilinear maps given above,

the multiplication example is symmetric but the determinant

example is not.

77

Remarks 1.

If

f(x)

=

(f

(x) ,f2 (x) ,. . . ,fm(x) )

straightforward to prove that the

rth

is

C

(See Remark 2,

2. An n-multilinear map

n

the

n

is

in

Rn

then it

is

precisely when all

f^

exist and are

§2.4.)

ExEx...xE->-R

covariant tensor of rank tensor of rank

x

(r 5 1)

mixed partial derivatives of all

continuous.

3.

f

where

on

E.

When

is specified by

m11

is sometimes E = Rm

called a

a covariant

coefficients.

In the discussion of multilinear maps we have avoided mentioning

topologies for

M

(E^ x E^ x

manage without these since,

...

x E

;F) .

Strictly speaking we can

as we showed,

the

spaces are just

alternative versions of the complicated spaces of continuous

linear maps

for which at least in principle we already have topologies derived from norms:

see §2.3.

Nevertheless,

it is useful to see how to make

into a normed linear space explicitly. continuous linear

M

n

By analogy with the theory for

(= 1-multilinear) maps,

it can be shown easily that

an n-multilinear map M :

E. x E„ x 12

. ..

x E

-> F

n

is continuous if and only if there exists a constant ||m(x)||

$ K

\\xl\\

1 lx2l

=

least such

for all

K, 1

-

...

x =

-

1

(x^,x^,...,x ) x

|| n1 1

=

M||

= sup{

with

If we then take

1M |

|m(x)I

1111

M

n

(E

1

= 1

x E? x 2

f°r

...

i =

x E

n

1,2,

;F)

.

this is in a sense the most reasonable norm to choose,

natural bisection between

78

such that

to be the

i. e.

then this does define a norm for more,

1

K

. . . ,n)

What

is

since the

and the corresponding space of linear maps

^(E1’L(E2> • • •

etc.)

is

then not only a linear isomorphism but is also

norm-preserving.

Properties of higher derivatives Corresponding to the properties of

the derivative mentioned in §2.5 we

have the following properties of higher derivatives:

(i)

Linear combinations of

Cr

Dr(af + gg)

(.11)

Linear maps

because

(in)

DL(x)

L

= L

are

B

Cr,

= aDrf + gDrg

C

,

for every

Bilinear maps

maps are

are

with x,

C

,

.

D L = 0

i.e.

with

and

for all

DL(x)

DB

r :> 2.

This is

does not vary with

as given in §2.5

x.

and

D B

given by D2B(x) ((h1,h2) , (k1,k2))

To see this, 2 D B(x)

observe that

= D(DB)(x)

= DB

D2B(x)(h,k)

=

= B(h1,k2)

DB(x)

is linear in

for all

x.

(D2B(x).h).k

DrB = 0

2 D B(x)

Since

for all

x

This means

.

and so that

by definition

= DB(h).k as claimed.

+ B^,!^)

by the above does not vary with

x

we have

r 5 3. oo

In a similar way it follows that n-multilinear maps

M

are

C

,

and D^MCx) (h,k, . . . ,p)

where there are

n

permutations

0

of

follows

D^M = 0

that

- £ M(h^^ ,k o

entries

h,k,...,p

{l,2,...,n}. for all

• »Pp)

and

the sum is over all

Since this

r £ n+1

is

independent of

x

it

.

79

(iv)

The cartesian product of two

Cr

CV,

maps is

and all derivatives

just operate co-ordinate-wise.

(v)

The composition of two

Cr

maps

is

Chain Rule together with induction on D(g* f)(x) so if

f

and

g

are

= Dg(f(x)) C

r

then

•

Cr.

r.

This comes from the

We have

Df(x)

Df

and

Dg

are

induction hypothesis

(that the result holds for

Dg(f(x))

function of

is a

C

x.

C

and the

r-1)

ensures that

Then we use the bilmearily of

the map L(E,F)

x L (F ,G) -* L(E,G)

:

(A,B)

B-A

and the Chain Rule and induction hypothesis again to see that C

r—1

and so

g*f

. is

C

D(g*f)

is

r

There is a formula for

D

(g*f)

in terms of derivatives of

f

and

g,

but it is cumbersome and not worth trying to memorize.

2.7 If x

GERMS AND JETS f

and

g

are two maps which agree on some neighbourhood of a point

then all their derivatives at that point are the same.

Therefore if

we are interested in trying to deduce the local behaviour of a map from information about its derivatives at

x

the precise nature of the map away from consider any map coinciding with

f

we need not be concerned with x

but could equally well

on some neighbourhood of

x.

This leads naturally to the idea of the germ of a map Let and

80

g

T :

be a topological space and let V ■+ S

be maps with domains

U,V

S

be any set.

open sets

in

Let T,

f

: U

and suppose

S x

lies

in

U n V.

Then

if there exists

f|w

such that

f

and

are said to be germ-equivalent at

g

some open neighbourhood = g|w,

i.e.

f

and

W

g

of

x

u r\ V

lying inside

coincide on

W.

x

This is an

equivalence relation on the set of all maps defined on neighbourhoods of x

in

T

and with values

germs of maps at

x.

in

If

S,

S

and the equivalence classes are called

is a topological

consider germs of continuous maps, E,F

we can consider germs at

these germs by

x

and if of

Cr

space also then we can

S,T

are normed linear spaces

maps.

We denote the set of

E^CEjF).

In any of these cases,

if

S

is

a linear space we can define

pointwise addition and scalar multiplication of maps from a given domain into at

S, x

and this goes through to the germ level to give the set of germs

the structure of a linear space.

If

S

also has a multiplicative

structure which interacts reasonably with addition and scalar multiplication then this,

too, will go through to a similar structure on

the set of germs at

In particular,

x.

if

is a ring or an algebra

S

(whose definitions we will not go into here)

then the set of germs at

x

becomes a ring or an algebra. The cases which will mainly interest us are those for which and in

S = R™ . Rn,

Taking

x

without loss of generality to be the origin

we see that the set

E^(Rn,Rm)

does and linear combinations of §2.6).

In the case

algebra,

and so

properties two

C

(iii)

m = 1

E^(Rn,R) and

functions

(v)

is

C

T = Rn

C

maps are

the codomain forms a ring

of

forms a linear space since

R

C

(property

(i)

R™

in

is a ring and even an

(algebra):

here we use

§2.6 to show that the pointwise product of

.

The ring

EQ(R

,R)

of

C

germs

is often CO

denoted simply

En

or

E(n).

When studying local properties of

C

81

functions of

n

elements of

E

variables we will really be studying properties of

n

0

CO

As we have said,

if two

their derivatives at

x

C

maps are germ-equivalent at

are the same.

about the derivatives at

map representing that germ.

rather than of a specific

The basic question then is:

does knowledge of all the derivatives of a germ at about the germ itself? derivatives

at

then all

This means that we can talk

of a given germ,

x

x

x

to what extent

provide knowledge

Can the germ be reconstructed,

given all its

x?

The most elementary result in this direction is the Mean Value Theorem (§2.5) which for

f

:

R -+ R

f (x+h)

E,

for some

states that

lying between

x

which is not a derivative at f(x+h)

+ hf' (O

= f (x)

= f(x)

and x.

h.

Of course,

this

f'(5)

Therefore

+ hf'(x)

+ error term.

Further manipulations with the Mean Value Theorem (as book on calculus)

involves

in any standard

show that the error term can be expressed as

h2 — f" (52)

f°r some

?2

lyi-n§ between

x

involves a derivative elsewhere than at working with derivatives at

x

and x,

x+h.

Again,

this

and so if we insist on

we must write

,2 f(x+h)

= f(x)

+ hf'(x)

+ — f"(x)

+ error term.

Once more the Mean Value Theorem can be used to show that the error term will have the form

h3

-jr f"(£3)

for some

E,^

between

x

and

x+h.

This process of pushing the error term further to right can be continued indefinitely,

although there is no reason to believe that the error term

will become any smaller as this happens.

82

Indeed,

it can easily get larger.

is described by Taylor's Theorem, which says

The true state of affairs that if we write

f(x+h)

= f(x)

then the error term,

, + —y f" (x) ^•

+ hf'(x)

+

...

or remainder term, has . n+1

,

, n . . + —r f n (x) n•

+ error term

the form

,.

V«-^(n+1) the error term

Rn(h)

has the form

—5—r Dn+1f(5)(h,h,...,h) (n+1) .

for some point

K

between

x

and

x+h

83

Tl

Remember that here

E x E x and so

...

Drf (x) (h,h,. .. ,h)

euclidean space Drf(x)(h ,h

Rn h)

l

•

D f(x)

•

is an r-multilinear map x E -> R

is a real number.

1

to

E

is

we can for purposes of calculations write as

3x.3x.. . . 3x i J P

(x)

h.h....h

1 J

p

where the sum is over all ordered r-tuples from

Recall also that when

i,j,...,p

of the integers

n.

The proof of the general version of the given the one-dimensional version. F(t) and apply the theorem to F'(t)

theorem is surprizingly simple,

We write

= f(x + th)

F

at

t = 0.

First of all

= DF(t).1 = Df(x + th).h

by the Chain Rule,

and then since the map

is linear for each fixed F"(t)

h

L(E,R) -* R

taking

A

to

A.h

the Chain Rule applied again gives

= D^f(x + th)(h.h)

and in general F^ (t)

= Drf (x + th) (h.h

Now writing

F(1)

= F(0)

stated for If it is

+ F' (0)

+ \y F"(0) ^•

+

...

+ ^- F(n) (0) n•

+

F(n+1) (£)

f. the case that

R

(h)

does approach zero as

we are, by definition of convergence for infinite series,

84

, v,n+1) •

n

increases then entitled to write

f(x+h)

+ Df(x).h+ JJ- D2f(x)(h,h)

= f(x)

+

...

CO

l

=

~r Dnf(x)(h,h,...,h)

,

n=0 n* where

D f

means

f.

The infinite series on the right is called the

Taylor series for (the germ of)

f

at

x.

This definition can be generalized directly to maps the series

itself

(now a series of elements in

Mean Value Theorem.

is

C

F)

f

:

E -+ F,

since

does not depend on the

If

f

the Taylor series always exists, but

(1)

it may diverge,

or

(2)

it may converge, but to something other than f(x+h) .

If neither of these happens, for all

h

at

For

x.

in some neighbourhood of E = R

,

F = R

for sufficiently small co-ordinates

and the Taylor series converges to

fu ,

then

this means that

f

is said to be analytic

f^(x+h)

can be expressed

as a convergent power-series in the

for each component

sometimes called class Analytic maps

h

x

f(x+h)

Cw

f^

of

f.

Analyticity is

.

are convenient to work with,

since we can invoke the

whole machinery of differentiation of power-series term by term, inversion of power series, any analysis text.

etc.,

Furthermore,

of which details can be found in almost powerful results from the theory of

complex functions can also be used.

What is perhaps even more

significant for practical purposes is the fact that if we replace

f(x+h)

by f(x)

+ Df(x).h + yr °2f(x)(h,h)

which is a polynomial in

h,

+

...

+ yr Dnf(x)(h,h,...,h)

then we can be confident that by choosing

n

85

large enough the error in taking this approximation will be insignificantly small. Despite these enticing advantages of analytic functions,

implicitly

relied upon in much mathematical modelling in physics, engineering and so on,

there are two serious objections - one practical and one

philosophical - to working with analytic functions.

The practical

objection is that functions do arise in real life which look harmless CO

but are in fact not analytic even though they are example is the function

f (x)

f

=

:

R -> R

^

,

x > 0

,

This is not analytic at induction)

0,

0

A standard

,

x ■ Rn

(f(u,v),v)

(I =

Inverse Function Theorem 0

:

with

has matrix

which is non-singular

germ at

F

Rn = Rm x Rn m

Hence

f

L(u,v)

is right equivalent to

is

A,

proving

Corollary 1.

Proof of Corollary 2 Again we suppose

x = 0

Rn, f(x) = 0

in

r

co-ordinates such that non-singular matrix and Rn x Rm n map

G

:

A

has matrix C

is an

(m-n)

R

R

DG(0)

M

and we choose

where now

v. L. x n

with corresponding co-ordinates

B

matrix. (x,y)

is an We write

K

as

and this time define a

= f(x)

+

(0,y)

.

has matrix

B

0

C

I

which is non-singular and so by the Inverse Function Theorem a diffeomorphism.

n x n

by G(x,y)

Then

g

Rm

in \

Let

3

be the germ at

0

of

M*G 1

G

is locally

•

97

Then M(x,0)

= gG(x,0) = Bf(x)

giving

B*f = A

equivalent to

since

A,

M(x,0)

=

Bx (Cx ^ V.

= Ax.

proving Corollary 2.

Hence

f

is left

See Figure 16.

Rm" y

Rm JL—

/TV

,

Rn x

0

Rm

f,

R

J X d&

/fR"

Figure 16 The condition that zero,

i.e.

A

is injective means that the kernel of

the only n-vector

u

satisfying

Au = 0

is

A

is

the zero vector.

n This means that the only linear combination J

)

u.c.

L,

of the columns

l—i

c. —l

i=l

of

A

(as a matrix)

words

the columns of

which is zero is the one with all u. A

are Z■{■nearly

'independent.

is surjective precisely when the rows of The

rank

= 0,

l

A

Correspondingly,

A

are linearly independent.

of a matrix is the largest number of linearly independent rows,

or alternatively,

columns

(the answer is the same).

Thus

corollaries above can be summarized as follows:

If

Df(x)

has maximum rank then

Zike the linear map

98

or in other

Df(x)

near

f 0.

Zooks near

x

the two

Remarks In infinite dimensions we cannot talk about ranks,

1.

the first formulation of the corollaries.

so we must stick to

We also have to add in an

extra condition to ensure that the splitting of

E

suitable

then the proofs are the

same as

'co-ordinates'

(x,y)

is

still valid:

in the finite-dimensional case.

Explicitly,

that in Corollary 1 we should be able to split © E2 with

(see page 42)

E^ = ker A

and

where E^

E^,E0

taken by

or

E

A

takes

A

isomorphic ally onto

F^.

2.

E

F

r>n“in

R

nm

into

or

F^ $ F^

is finite-dimensional

is finite-dimensional in Corollary 2.

In the context of Corollary 1 we can define a 2

F

F,

E

These splitting conditions

can be shown to be automatically satisfied when in Corollary 1 or

the condition is

are closed linear subspaces of

isomorphically onto

E

into two

into a direct sum

similarly in Corollary 2 we should be able to split where

F

C

map

(germ)

"t

-> R

by (y)

= first co-ordinate of y(- B "*"Cy,y)

,

where the inspiration for the expression on the right is that the linear subspace of

Rn

consisting of all elements of the form

precisely the kernel of

A.

It then follows

(- B

Cy,y)

is

that

f((y) »y) = fy(- B lcy>y) becaus-e

y

preserves the y co-ordinate

(because

L

and

F

do) ,

= A(- B_1Cy,y)

=

by design.

0

This version of Corollary 1 is called the

Implicit Function

Theorem, since it says that if f is

a

Cr

:

Rn = Rm x Rn_m -+ Rm

map with

f(0)

=0

and such that the derivative of

f

99

restricted to the first factor is an isomorphism (represented by the matrix

B

above) ,

then there exists a

Cr

map

Rm

satisfying

f (4> (y) >y) = 0 for all f = 0

y

in some neighbourhood of the origin, or in other words putting

gives the first co-ordinate as an implicit function

second co-ordinate.

Note that when

n = 2, m = 1

and the derivative condition is simply that



Rm

Rn.

DEFINITION

A singular point of

f

maximal rank3 i.e. where The germ of

100

f

is a point Df(x)

x

at which

Df(x)

does not have

is neither injective nor surjective.

at a singular point is called a singularity.

be

Thus at a singular point the derivative may not give a good local qualitative picture of the map itself.

Remark The term

singularity

is used in many different fields

of different things - such as a point where not continuous,

or not defined.

f

to mean a variety

is not differentiable,

or

It may also refer to a region of space

with bizarre geometric properties,

such as a

'black hole'

in cosmology.

Here we will keep the word to mean the germ of a differentiable map at a singular point.

EXAMPLES of singularities 1.

In the case

n = m = 1

the derivative

Df(x)

is a linear map

which must either be surjective or be the zero map. point of

f

R ->■ R

:

is a point

x

where

f' (x)

: R

point or stationary point.

For example,

has

and nowhere else.

a singularity at

2.

x = 0

=

f

n,

i.e.

to functions

the linear map

Rn -*■ R

f

:

R.

U

R

'df

8f

given by

f (x)

m = 1

= x

for

Df (x)

is

(=1)

is

matrix

8f

ax * ax ’ ’‘‘ ’ ax 12

v.

x n

1

R

called a critical

0,

Here the derivative

represented by the

->

Thus a singular

The previous example generalizes immediately to the case

any

R

ny

and so the only way in which it can fail to have maximal rank 3f

by the vanishing of all the partial derivatives

7—-

.

Again,

this is

i called a critical point or stationary point of is

f

:

Rn

->

R

singularity at is

given by 0

in

Rn

f(x)

= x^

2

+

2

+

but nowhere else.

in a certain sense the most

'nonsingular'

f.

...

+ x

A standard example

2

;

this has a

As we shall see later,

this

type of singularity.

101

A constant map

3.

Rn)

is

Df (x)

f

Rn

:

Rm

(i.e.

f (x)

= c

is the zero linear map

Rn

Rn -* R™

is crushed by

f

Rm

in

the most singular kind of map imaginable.

the structure of 4.

f

for each

for all

x

in

Here the derivative x,

and the whole of

into a single point of

R .

As a geometrically more interesting kind of singularity consider

R2 -* R2

:

given by 3

f(x1,x2) = (xlfx2

Here

Df(x)

+ x1x2)

is represented by the

.

2x2

matrix

3x22 +

and so the rank is less than maximal precisely when

3x22 + x^ = 0

which is the equation of a parabola in the domain be described pictorially as

R

2

.

The whole map can

in Figure 17.

Critical points of real-valued functions We will now return to the case when one real function of i.e.

102

n

m = 1,

in other words when

f

is

variables and singular points mean critical points,

points at which all the partial derivatives of

f

vanish.

To understand how

f

behave s near a critical point,

look at the next available its

second derivative.

'approximate'

Recall

it is natural to

information about

f,

that this second derivative

namely

D2f(x)

can

be regarded as a symmetic bilinear map D2f(x) or as

a quadratic form,

Hf(x)

of

f

at

x.

:

Rn

x

Rn + R

represented by the Hessian matrix

32f

=

(x)

9x.3x. 1 J

From experience with the Inverse Function Theorem it seems

reasonable that we will be able to extract the most local information about f

from its

second derivative when this Hessian matrix is non-singular.

DEFINITION

A critical point

x

of

f

:

U ->

R

is called degenerate ov non-degenerate

according to whether the determinant of the Hessian matrix of

at

f

x

does or does not vanish.

Remarks 1.

The Hessian matrix itself depends on a choice of co-ordinates

but the vanishing or otherwise of its determinant does not.

Rn,

in

This

is easy

to verify by applying a co-ordinate change and then using the Chain Rule twice.

However,

the real reason for this is that the vanishing or

non-vanishing of the determinant of

H^(x)

corresponds to the

. .

degeneracy or non-degeneracy of the bilinear map An arbitrary bilinear map if there exists

some

u

B in

: Rn

Rn

x

Rn

R

is

2

D f(x).

said to be degenerate

with the property that

B(u,v)

= 0

every

v

in

Rn

;

point

x

of

f

is degenerate or non-degenerate according to whether

otherwise it is non-degenerate.

Thus

for

the critical

103

2

D f(x) 2.

is a degenerate or non-degenerate bilinear map.

In view of the previous remark, we can extend the theory of critical

points to functions on any normed linear space critical point of

f

when

Df(x)

R

: E ->

is

E.

The point

x

the zero linear map,

is a and is

degenerate or non-degenerate according to the degeneracy or

2 non-degeneracy of the bilinear map

D f(x)

:

E x E ->

R.

The Inverse Function Theorem allowed us to change co-ordinates to bring f

locally into linear form,

as possible.

provided the derivative was as well-behaved

We might next hope for a theorem saying that even at a

critical point we can change co-ordinates agreeable standard form, non-degenerate.

(b.

f

locally into some

provided that the critical point is

There is

after Marston Morse

to bring

such a theorem.

1892

It is called the Morse Lemma,

) who gave it as part of a general theory

of the structure of real-valued functions,

a theory which has had far-

reaching implications in differential topology.

Proofs may be found in

many books,

or Milnor

such as Golubitsky and Guillemin

[42]

[[82].

THEOREM (Morse Lemma)

Suppose f

at

p

is a non-degenerate critical point of is right equivalent to the germ at

p

f.

Then the germ of

of a function

0

cf>

of

the form \

+

> • • • )

where each

e.

i

is

It is clear that origin,

104

since

D (x)

± 1

=

...

+ e x n n

2

.

does have a non-degenerate critical point at the ,2e2x2 ,. . . ,2cnXn)

(as a

1 x n

matrix)

which vanishes at

x - 0,

and

H^(0)

is the

n x n

diagonal matrix

2f

2e,

2e

which is non-singular. example,

n

The Morse Lemma says that this is the archetypal

every other function near a non-degenerate critical point being

reducible to one of this type by a local change of co-ordinates. The number of

e.

which are

is called the i-ndex of the critical

- 1

1

point

p.

It depends only on

f

near

way of choosing new co-ordinates;

it is

p,

and not on the particular

the number of negative

2 eigenvalues of

H^(p)

or of

D f(p)

as a linear map

is

we can rearrange co-ordinates so that

Rn -* L(Rn,R) = Rn . If the index of has

p

k

(apart from the constant

f(p))

the form \

/*

2

(x-^ ,X2 , • • * , x^) V

X1

+

. . .

+ x

2

2

2

, n-k

Xn-k+l

...

+

X

n

This allows us to gain a good geometric picture of the behaviour of near

0,

i.e.

of

index

(including

f

cj>

near

k = 0)

p. and

There are n + 1

n + 1

possible values of the

correspondly different types of

non-degenerate critical point. n = 1

n = 2

k = 0

Local form

x

minimum

k = 1

Local form

- x

maximum

k = 0

Local

form

x-^

k = 1

Local form

x^

k = 2

Local form

- x^

2

+ x2

2

2

:

2

saddle point

- x2

2

- x2

minimum

2

:

maximum

105

See Figure 18 for pictures in the case

n = 2.

Note that a saddle

point can be thought of as a minimum in one direction coupled with a maximum in another.

In general there will be

saddle point corresponding to

(n-1)

different types of

k = 1,2,...,n-1.

Remarks 1. An immediate deduction from these local forms non-degenerate critical point is iisolated, containing no other critical point,

i.e.

is that each lies

in some neighbourhood

degenerate or otherwise.

2. The Morse Lemma has an infinite-dimensional version,

extremely useful

in calculus of variations where it is necessary to consider the behaviour of a function defined on a normed linear space themselves functions. write the

(rearranged) x ^ f(p)

where

P

canonical form for +

||(I-P)x||^ -

E

Rn -> Rn

.

above form where

P

whose elements are

In finite dimensions we could (J>

as

||Px||^

k

co-ordinates and

I

denotes the

The infinite-dimensional Morse Lemma states

is a Hilbert space

degenerate critical point

106

[92].

is projection onto the last

identity map provided

See Palais

E

p

(page 53),

the function

that,

in a neighbourhood of a non¬ f

can be converted into the

now denotes a perpendicular projection onto some

linear subspace of

E.

There do exist generalizations of this to Banach

spaces which are not Hilbert spaces:

see Tromba

|44] .

Further study of degenerate critical points At a degenerate critical point

p

singular, which is equivalent matrix has non-zero kernel, with

Hf(p).u = 0.

of

f

the Hessian matrix

H

to saying that as a linear map

i.e.

However,

(p)

is

R° -* Rn

there exists at least one direction

the u

it would be reasonable to hope that if we

keep perpendicular to such directions we could still apply a kind of Morse Lemma giving a standard quadratic form in some co-ordinates, completely degenerate

(i.e.

the remaining variables. choose co-ordinates non-degenerate does exist.

'Morse'

(Splitting Lemma,

conveniently

[45^ •

or Gromoll-Meyer Lemma)

is a degenerate critical point of

is right equivalent to the germ at

0

(xi5X2, . . . ,Xn) B- f (p)

e.

is

±1

+

and

Then the germ of

f.

of a function

/

where each

allowing us to

It was first published in an infinite-dimensional context by

THEOREM

p

Such a generalized Morse Lemma,

part and a completely degenerate part,

and Meyer

p

function of

so as to split the function locally into the sum of a

Gromoll

Suppose

degenerate in all directions)

leaving a

e,x, 11

ip

+...+£ x

2

r r

at

of the form

f

k

2

f

+

rj,

[ r+1

is completely degenerate,

5 nj

i.e.

i

D2ip(0)

= 0

(as well as

Dip(0)

= 0).

Remark In the infinite-dimensional version when

E

is a Hilbert space

H

we let

2 K

be the kernel of the second derivative

H -> L(H,R)

= H*

.

D f(p)

(In fact for a Hilbert space

regarded as a linear map H

= H,

in much the same

107

way that

Rn" = Rn;

seepage 247.)

so that there is only a 2 D f(p)

the image of

We assume that

K

is finite-dimensional,

'finite-dimensional amount' a

is a closed subspace of

of degeneracy,

, (a technical necessity).

H

Each point

x

can be regarded as having co-ordinates

belongs to

K

and

w

is perpendicular to

K.

and that

(w,y)

where

y

Then the standard form

given by the Lemma is x b- f(p)

where this time perpendicular to element of

+

| | (I-P)w| |2 -

P

| |Pw| |2 +

(y)

is a projection inside the space of vectors K,

L(K,K )

and

(Jj

satisfies

DiJj(O)

or as a bilinear map

In view of the Gromoll-Meyer Lemma,

= 0

and

K x K -v R

2

D i[j(0)

w =0

as an

.

the problem of studying the

behaviour of degenerate critical points has been narrowed down to the study of critical points at which the second derivative vanishes entirely, i.e.

points at which the 2-jet vanishes

f(p)).

(ignoring the initial constant

It would be pleasant if there were some kind of

'non-degeneracy'

condition on the third derivative in the presence of which we could again convert the function locally into some standard form by a sort of third degree Morse Lemma.

Essentially this is the case, but the details of the

theory, which generalizes to functions with vanishing 3-jets, are rather more subtle than one might at first suspect.

4-jets,

Indeed,

etc.,

it is

only in recent years with the work of Thom, Mather, Arnol'd and others that the classification of degenerate critical points under various conditions of relative non-degeneracy has become properly understood at all. The problem can be stated in two parts: (1) To what extent does the first non-vanishing k-jet of

108

f

determine the local behaviour of

f ?

(2)

Can the first non-vanishing k-jet homogeneous polynomial degree

k)

(which is a

x1,x2,...,x

in

of

be put into some normal form?

The second problem is one of algebraic geometry, or

less easily for a few low values of

k - 2

k

and

and can be solved more

n.

For example, when

we have the classification of quadratic forms

into standard non¬

degenerate types

ex If

2

+ e x„ 22

2

+

...

+ e x n n

which appeared in the Morse Lemma,

2 elXl

2 + e2X2

2

(e. = ± 1) x

and degenerate types

2 +

•••

(ei = ± 1, 0 $ r < n)

+ erxr

which featured in the Gromoll-Meyer Lemma. cubic form in coincide)

x^,x2

factorizes

When

k = 3

and

n = 2

into one or three real factors

a

(which may

giving without much effort the various standard forms 3

X1

3

+ X2 ’ X1

The first problem is

3

„

2

2

3

~ 3xix2 ’ xl x2’ X1

that of detevmi-naoy of the k-jet

(see §2.7).

Without going into the depths of the theory, we will give the result in the form of a finite procedure for testing the determinacy of the first non-vanishing jet. We may as well suppose that we are working at the origin in

Rn,

and

we shall regard the first non-vanishing jet of a function germ as being a homogeneous polynomial of the appropriate degree a k-jet,

and for

i = l,2,...,n

let

d^

denote

k. 9E

,

Let

E,

be such

itself either

i zero or a homogeneous polynomial of degree

k - 1.

109

DETERMINACY TEST.

The jet

E,

every homogeneous polynomial

as above -is determined if and only if of degree

q

k + 1

in the

can he

x^

written as q = q1d1 + q2d2 + ... +

where the

are homogeneous polynomials of degree

2.

Since the space of homogeneous polynomials of degree dimensional real linear space of dimension

ml

X1

/-a 1,-4(n-1);k.

is a finite-

k

(the elements

m

m2

x

X2

n

with

m, + m~ + 12

...

form a has is),

+ m = k n

the

Determinacy Test reduces to a finite combinatorial problem.

EXAMPLES 1. The k-jet of a constant function is not determined since each However, 2.

If

1-jet

- 0.

this fact is obvious anyway.

Df(0)

E,

d.

of

/ 0 f

(i.e.

0

is not a critical point of

is determined since at least one of the

is a non-zero number and then every polynomial can be written as

f)

d^

q

then the d^

of degree

times a polynomial of degree 2.

(say

d^)

k+1 = 2

However, we

already knew this fact as a consequence of the Inverse Function Theorem in §2.8. 3. If

Df(0)

= 0

then if

A

denotes the Hessian matrix

H^(0)

we have

n 1.

\

=

1 If

A

is non-singular

there is a matrix

(i.e.

B = A

A. .

x.

^

J

j-1 0

is a non-degenerate critical point of

such that n

x. J

110

=

y

i=l

B. .

d.

J1

1

f)

Now every homogeneous polynomial

q

of degree

3

can be written as

n

(0)

= 0,

and then take

y = x(l +

(x))

1 /k

.

It is easy to make

argument rigorous.

Observe in constrast:5.

For

k > 1

the k-jet

(x

,x2) ►+

k

. . , is not determined,

since

x2

k+1

cannot be written as

q ll

3? - + q0 l2 3x.

The result is reasonable,

35

, k-1 — = q1kx1

3x,

since

1

x^

vanishes everywhere along a line

(the x2-axis) whereas no amount of local changes of co-ordinates

around

0

111

in

R“

x^ - x-^ dx2

will alter the fact that e.g.

also along a curve touching the More generally,

x2-axis at

vanishes

the origin.

it is easy to see both from the algebra and the

geometry that no k-jet of

f

:

Rn -> R

which is

independent of one or more

of the co-ordinates can possibly be determined.

6. The 3-jet (x1,x2) h- x1

is determined since

d^ = 3x^

3

2

- 3x^2

- 3x2

homogeneous polynomial of degree 4 q^d^ + q2d2

with

q

, q£

3 2 2 3 x^ x2> x^ x2 , x^x2

2

,

in

of degree

2

d2 = - 6x^x2 x^,x2

2.

can be written as

To see this,

are already multiples of

1

4

X1

2,

1

and every

d2

observe that

and that

,

= 3 X1 dl " 6 XlX2d2 4

x„ 2

1 2, i , = - -x x-. d, - — x,x„d„ 321 6122

However: 7.

The 3-jet (x

is not determined, of form

d^

and

d2

,x2) ^ x^ x2

since

d^ = 2x^x2, d2 = x^

would have

q^d^ + ^2d2

'

x^

2

as a factor.

and so every combination Thus

x2

4 is not of the

This non-determinacy can be understood geometrically

2

as follows:

x^ x2

vanishes along two lines

cutting at right angles, whereas along the x^-axis only.

2 5 x^ x2 + x2 ,

axes)

for example, vanishes

No local co-ordinate changes

one of these configurations into the other.

112

(the co-ordinate

in

R

2

can convert

So far we have considered determinacy only for homogeneous k-jets, which materialize as the first non-vanishing jet at germ.

0

of some function

It is important also to be able to test determinacy of non-

homogeneous jets, since it frequently happens that although the first non-vanishing jet of

f

may not be determined, the addition of the next

term in the Taylor series of

f

gives a jet which then is determined.

The necessary and sufficient test for determinacy of non-homogeneous jets is a little cumbersome, and in many cases it is enough to work with two tests, one necessary and one sufficient, which do not quite meet in the middle. Let

£

For full details see Poston and Stewart

[lOl] .

be any (possibly non-homogeneous) k-jet, and define

3 E,

d^ =

i as before. any degree

In this case $ k - 1.

d^

Let the minimum degree of a polynomial denote the

lowest degree of all its terms. 2

may be a non-homogeneous polynomial of

For example, the minimum degree of

3 2. X2X3 + xp x3 1S four*

Sufficient test.

If every homogeneous -polynomial

q

of degree

k + 1

can be written as n q =

l

q^i + r

i=l where the

q^

cere polynomials of minimum degree at least

a polynomial of minimum degree at least

k + 2,

then

5

23

and

r

is

is determined.

This is similar to the earlier test (for homogeneous jets), but we now allow the > k + 1

q^

to be non-homogeneous and discard all terms of degree

which arise.

113

EXAMPLE We saw previously that

If now we let

n

(x^,X2) b- x^ X2

denote the 4-jet

x

is not a determined 3-jet.

(x^,X2) b- x^ X2 + x^

n

1

is determined

(here

r = 0).

Thus

5

have

Necessary test.

If

of degree

can he written as

of minimum degree

as a factor and

X2

4

(*)

where the

q^

are polynomials

is as before.

r

Clearly the sufficient test implies the necessary test, However,

x^ X2

is determined then every homogeneous polynomial

£

and

1

d^

the undetermined 3-jet

has been made into a determined 4-jet by adding on

k + 1

we find that

5

and other homogeneous polynomials of degree so

4

as

it should do.

there is a small gap between the two through which jets can

easily fall.

EXAMPLE (Siersma's example

[l 1 ^ 3

Let

5

be the 4-jet

d2 = 3x^X2

2

.

.

.

so we suspect that

£

minimum degree

We have

d^ = 3x^

3 + x^

2.

5

6)

with

Thus the Sufficient Test fails

and

To prove this we must

q^d^ + ^2^2

(in particular we see

ea°h of 5

2 = x? d^ - x^^)

and so

does not fail the Necessary Test and we cannot be sure that it is not

114

,

here we find that any homogeneous polynomial

can be written as 1

5

(m°dulo terms of degree

may not be determined.

apply the Necessary Test: 5

.

.

q^d-^ + ^2^2

each of minimum degree

of degree

+ x.^2

A little combinatorial work shows there is no way in which

can be written as q^,q2

2

3

(x^,X2) b- x^

E,

determined: determined,

it has fallen between the two tests.

In fact this

E,

is

but to prove this needs more technicalities which we will not

discuss here. Interestingly,

polynomial of degree degree

7)

where

E,

for this 6

it turns out that every homogeneous

can be written as

qpq9

have minimum degree

regarded, as a 5-jet is determined, of

q^d-^ + ^2^2 2.

(modulo terms of

This means

E,

that

in other words that any function germ

the form 3

(x^,x2) ^

is right equivalent to

+ x^x^

E,.

3

+

(terms of degree

£ 6)

It is in showing that terms of degree

5

also make no difference that the subtlety of this example lies.

Remarks on the literature.

There are a great many books on linear algebra. A fairly random selection is Cullen [3l] , Halmos [49] , Mirsky [84] (in which linear spaces are called linear manifolds), Nomizu [91J , Shields The basic material on calculus in normed linear spaces can be found in Dieudonnd

[32] , Lang

[67]

or Loomis

and Sternberg

[70] ,

jE 14] .

and for euclidean

space in almost any text on advanced calculus such as Hoffman

[58],

Rudin [10^ or Spivak [l2| . See also Griffiths and Hilton [43] or Hirsch and Smale [55] for both linear algebra and calculus. Some references on singularity theory are Eells

[36] ,

Golubitsky and Guillemin

[42] t or the

survey by Arnol'd [9], Determinacy of jets is explored and explained thoroughly in Poston and Stewart [LOT]. Much of the content of this chapter is also contained in books on differential topology referred to at the end of Chapter 3,

and also in Field

[l60] .

115

3 Differentiable manifolds and maps

3.1

THE CONCEPT OF A DIFFERENTIABLE MANIFOLD

With the topology and calculus developed so far we have assembled a mathematical toolkit for dealing with

1.

differentiation in open subsets of euclidean spaces or, more generally,

2.

global structures which are not necessarily

The next step is of

in normed linear spaces

to unite these

two.

linear.

As a natural result of the synthesis

local linearity with global non-linearity we arrive at the definition

of a differentiable manifold. the definition of differentiability

Recall that in order to formulate at

x

for a continuous map

f

needed to use the linearity of of

x

in

U

and

f(x)

in

F.

:

U -> F,

where

U

is

locally, i.e.

E, F

Therefore if

S

open in

E,

in some neighbourhood

and

T

are two

topological spaces we can talk about differentiability of a map at a point

x

linear spaces context of all

S

provided that

in neighbourhoods of

topological spaces means

this more formally as Suppose

map.

Let

S, x

neighbourhood where

U'

that

f(x)

116

in

is

T

x

and and

S

:

S

S,

Now 'looks

'is homeomorphic to',

f

:

and suppose that

such that there is

S -> T x

like'

in the

so we can express

V

in

has

is

T

E.

a continuous

an open

a homeomorphism

an open subset of a normed linear space

has an open neighbourhood

T

look locally like normed

f(x).

are topological spaces and

in

T

f

follows.

be a point of U

S

we only

:

U -> U'

Suppose also

with a homeomorphism

ip : V -> V' assume

where

V’

is

an open subset of a normed linear space

for simplicity that

so that

f(U) U'

a

in a normed linear space r

a

.

The same

If we put in the further reasonable assumption

locally the same everywhere,

i.e.

that all

really just copies of the same normed linear space at

E

the definition of a manifold modelled on

E,

the

E^'s

are

we have arrived

E.

117

DEFINITION

A manifold modelled, on a normed linear space which admits a family

of open sets covering

{U^}

is homeomorphic with an open subset

Each

U ,

called a chart.

The collection

S. f

{(U

a

,