Mathematics.pdf

Timothy Gowers

MATHEMATICS

A Very Short Introduction

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UNIVERSITY PRESS

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Timothy Gowers 2002

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First published as a Vey Short Introduction 2002

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ISBN 978-0-19-285361-5

14 16 18 20 19 17 15

'eset by ReineCatch Ltd, Bungay, Sufolk

Printed in Great Britain by shford Colour Press Ltd, Gosport, Hampshire

Contents

Preface ix

List of diagrams xiii

Models 1

Numbers and abstraction 17

3 Proofs 35

Limits and ininiy 56

Dimension 70

Geometry 86

Estimates and approximations 112

Some frequently asked questions 126

Furher reading 139

Index 141

Preface

arly in the 20th century, the great mathematician David Hilbert

noticed that a number of important mathematical arguments were

structurally similar. In fact, he relized that at an appropriate level of

generliy they could be regarded as the sme. This observation, and

others like it, gave rise to a new branch of mathematics, and one of its

central concepts was named ater Hilbet. The notion of a Hilbert space

sheds light on so much of modern mathematics, from number theory to

quantum mechanics, that if you do not know at least the rudimens of

Hilbert space theory then you cannot laim to be a well-educated

mathematician.

What, then, is a Hilbert space? In a typical universiy mathematics

course it is defined as a complete inner-product space. Students

attending such a course are expected to know, rom previous courses,

that an inner-product space is a vector space equipped with an inner

product, and that a space is complete if every Cauchy sequence in it

converges. Of course, for those definitions to make sense, the students

also need to know the definitions of vector space, inner product,

Cauchy sequence and convergence. To give just one of them (not the

longest): a Cauchy sequence is a sequence

. . such that

x, x2, x.,

there exists an integer N such that for any

wo integers p and q greater than N the distance from

for every positive number

is at

x P

most::.

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