Aspects of Harmonic Analysis and Representation Theoryjean/nc-harmonic.pdf · For the groups listed above, we know that under some suitable restriction, we have Fourier inversion,

Aspects of Harmonic Analysis andRepresentation Theory

Jean Gallier and Jocelyn QuaintanceDepartment of Computer and Information Science

University of PennsylvaniaPhiladelphia, PA 19104, USAe-mail: [email protected]

c© Jean Gallier

August 12, 2019

Preface

The question that motivated writing this book is:

What is the Fourier transform?

We were quite surprised by how involved the answer is, and how much mathematics isneeded to answer it, from measure theory, integration theory, some functional analysis, tosome representation theory.

First we should be a little more precise about our question. We should ask two questions:

(1) What is the input domain of the Fourier transform?

(2) What is the output domain of the Fourier transform?

The answer to (1) is that the domain of the Fourier transform, denoted by F , is a setof functions on a group G. Now in order for the Fourier transform to be useful, it shouldbehave well with respect to convolution (denoted f ∗ g) on the set of functions on G, whichimplies that these functions should be integrable.

This leads to the first subtopic, which is what is integration on a group? The technicalanswer involves the Haar measure on a locally compact group. Thus, any serious effortto understand what the Fourier transform is entails learning a certain amount of measuretheory and integration theory, passing through versions of the Radon–Riesz theorem relatingRadon functionals and Borel measures, and culminating with the construction of the Haarmeasure. The two candidates for the domain of the Fourier transform are the spaces L1(G)and L2(G). Unfortunately, convolution and the Fourier transform are not necessarily definedfor functions in L2(G), so the domain of the Fourier transform is L1(G). Then the equationF(f ∗ g) = F(f)F(g) holds, as desired. If G is a compact group, L2(G) is a suitable (andbetter) domain.

The answer to Question (2) is more complicated, and depends heavily on whether thegroup G is commutative or not. The answer is much simpler if G is commutative. In bothcases, the output domain of the Fourier transform should be a set of functions from a spaceY to a space Z.

3

4

If G is commutative, then we can pick Z = C. However, the space Y is rarely equal to G(except when G = R). It turns out that a good theory (which means that it covers all casesalready known) is obtained by picking Y to be the group Ĝ, the Pontrjagin dual of G, whichconsists of the characters of the group G. A character of G is a continuous homomorphismχ : G→ U(1) from G to the group of complex numbers of absolute value 1. For any functionf ∈ L1(G), the Fourier transform F(f) of f is then a function

F(f) : Ĝ→ C.

In general, Ĝ is completely different from G, and this creates problems. For the familiarcases, G = T ∼= U(1), G = Z, G = R, and G = Z/nZ, the characters are well known, namelyT̂ = Z, Ẑ = T, R̂ = R, and Ẑ/nZ = Z/nZ. The case G = Z/nZ corresponds to the discreteFourier transform.

For the groups listed above, we know that under some suitable restriction, we have Fourierinversion, which means that there is some transform F (called Fourier cotransform) suchthat

f = F(F(f)). (∗)

We have to be a bit careful because the domain of F is L1(Ĝ), and not L1(G), are they areusually very different beause in general G and Ĝ are not isomorphic. Then (assuming that it

makes sense), F(F(f)) is a function with domain ̂̂G, so there seems no hope, except in veryspecial cases such as G = R, that (∗) could hold. Fortunately, Pontrjagin duality assertsthat G and

̂̂G are isomorphic, so (∗) holds (under suitable conditions) in the form

f = F(F(f)) ◦ η,

where η : G→ ̂̂G is a canonical isomorphism.If G is a commutative abelian group, there is a beautiful and well understood theory of the

Fourier transform based on results of Gelfand, Pontrjagin, and André Weil. In particular,even though the Fourier transform is not defined on L2(G) in general, for any function

f ∈ L1(G) ∩ L2(G), we have F(f) ∈ L2(Ĝ), and by Plancherel’s theorem, the Fouriertransform extends in a unique way to an isometric isomorphism between L2(G) and L2(Ĝ)

(see Section 10.8). Furthermore, if we identify G and̂̂G by Pontrjagin duality, then F and

F are mutual inverses (see Section 10.9).If G is not commutative, things are a lot tougher. Characters no longer provide a

good input domain, and instead one has to turn to unitary representations . A unitaryrepresentation is a homomorphism U : G → U(H) satisfying a certain continuity property,where U(H) is the group of unitary operators on the Hilbert space H. Then Ĝ is the set ofequivalence classes of irreducible unitary representations of G, but it is no longer a group.

If G is compact, an important theorem due to Peter and Weyl gives a nice decomposi-tion of L2(G) as a Hilbert sum of finite-dimensional matrix algebras corresponding to the

5

irreducible unitary representations of G (see Theorem 13.2 and Theorem 13.6). As a con-sequence, there is good notion of Fourier transform, such that the Fourier transform F(f)is a function with domain Ĝ, but its output domain is no longer C. Instead, it is a finite-dimensional hermitian space depending on the irreducible representation given as input (seeSection 13.4). In general, it is very difficult to find the irreducible representations of acompact group, so this Fourier transform does seem to be very useful in practice.

If the compact group is a Lie group, then the whole machinery of Lie algebras and Liegroups developed by Élie Cartan and Hermann Weyl involving weights and roots becomesavailable. In particular, if G is a connected semisimple Lie group, the finite-dimensionalirreducible representations are determined by highest weights. There is a beautiful andextensive theory of representations of semisimple Lie groups, and many books have beenwritten on the subject; see the end of Section 12.6 for some classical references.

A way to deal with noncommutativity due to Gelfand, is to work with pairs (G,K), whereK is a compact subgroup of G. Then, instead of working with functions on G, which is “toobig,” we work with functions on the homogeneous space G/K, the space of left cosets. Then,under certain assumptions on G and K, which makes (G,K) a Gelfand pair , it is possibleto consider a commutative algebra of functions on the set of double cosets KsK (s ∈ G), sothat some results from the commutative theory can be used (see Chapter 15). The domainof the Fourier transform is a set of functions called spherical functions , and this set happensto be homeomorphic to the set of characters on the commutative algebra mentioned above.There is a very nice theory of the Fourier transform and its inverse (see Section 15.5), buthow useful it is in practice remains to be seen.

Acknowledgement : Many thanks to the participants of the “underground” Tuesday meetings,Christine Allen-Blanchette, Carlos Esteves, Stephen Phillips, and João Sedoc, for catchingmistakes and for many helpful comments. We also thank Kostas Daniilidis for being asource of inspiration. Our debt to J. Dieudonné, G. Folland, A. Knapp, A.A. Kirillov, andW. Rudin, is enormous. Every result in this manuscript is found in one form or another intheir seminal books.

Contents

Contents 7

1 Introduction 11

2 Function Spaces Often Encountered 192.1 Spaces of Bounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Convergence: Pointwise, Uniform, Compact . . . . . . . . . . . . . . . . . . 212.3 Equicontinuous Sets of Continuous Functions . . . . . . . . . . . . . . . . . 292.4 Regulated Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Neighborhood Bases and Filters . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Topologies Defined by Semi-Norms; Fréchet Spaces . . . . . . . . . . . . . . 43

3 The Riemann Integral 473.1 Riemann Integral of a Continuous Function . . . . . . . . . . . . . . . . . . 483.2 The Riemann Integral of Regulated Functions . . . . . . . . . . . . . . . . . 53

4 Measure Theory; Basic Notions 574.1 σ-Algebras, Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Null Subsets and Properties Holding Almost Everywhere . . . . . . . . . . . 674.3 Construction of a Measure from an Outer Measure . . . . . . . . . . . . . . 694.4 The Lebesgue Measure on R . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Integration 775.1 Measurable Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Step Maps on a Measurable Space . . . . . . . . . . . . . . . . . . . . . . . 835.3 µ-Measurable Maps and µ-Step Maps . . . . . . . . . . . . . . . . . . . . . 855.4 The Integral of µ-Step Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5 Integrable Functions; the Spaces Lµ(X,A, F ) and Lµ(X,A, F ) . . . . . . . . 985.6 Fundamental Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . 1075.7 The Spaces Lpµ(X,A, F ) and Lpµ(X,A, F ); p = 1, 2,∞ . . . . . . . . . . . . . 1145.8 Products of Measure Spaces and Fubini’s Theorem . . . . . . . . . . . . . . 1215.9 The Lebesgue Measure in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Radon Measures on Locally Compact Spaces 129

7

8 CONTENTS

6.1 Positive Radon Functionals Induced by Borel Measures . . . . . . . . . . . . 1306.2 The Radon–Riesz Theorem and Positive Radon Functionals . . . . . . . . . 1366.3 Regular Borel Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.4 Complex and Real Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.5 Real Measures and the Hahn–Jordan Decomposition . . . . . . . . . . . . . 1476.6 Total Variation of a Radon Functional . . . . . . . . . . . . . . . . . . . . . 1506.7 The Radon–Riesz Theorem and Bounded Radon Functionals . . . . . . . . . 153

7 The Haar Measure and Convolution 1597.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.2 Existence of the Haar Measure; Preliminaries . . . . . . . . . . . . . . . . . 1717.3 Existence of the Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . 1777.4 Uniqueness of the Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . 1817.5 Examples of Haar Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.6 The Modular Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.7 More Examples of Haar Measures . . . . . . . . . . . . . . . . . . . . . . . . 1927.8 The Modulus of an Automorphism . . . . . . . . . . . . . . . . . . . . . . . 1937.9 Some Properties and Applications of the Haar Measure . . . . . . . . . . . . 1987.10 G-Invariant Measures on Homogeneous Spaces . . . . . . . . . . . . . . . . 2017.11 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2077.12 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8 The Fourier Transform and Cotransform on Tn, Zn, Rn 2278.1 Fourier Analysis on T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.2 The Fourier Transform and Cotransform on Tn and Zn . . . . . . . . . . . . 2448.3 The Fourier Transform and the Fourier Cotransform on R . . . . . . . . . . 2498.4 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2558.5 The Fourier Transform and the Fourier Cotransform on Rn . . . . . . . . . 2578.6 The Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2598.7 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . 2648.8 Pointwise Convergence of Fourier Series on T . . . . . . . . . . . . . . . . . 2658.9 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 2708.10 Fourier’s Life; a Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . 272

9 Normed Algebras and Spectral Theory 2759.1 Normed Algebras, Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . 2799.2 Two Algebra Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849.3 Spectrum, Characters, Gelfand Transform, I . . . . . . . . . . . . . . . . . . 2889.4 Spectrum, Characters, II; For a Banach Algebra . . . . . . . . . . . . . . . . 2949.5 Gelfand Transform, II; For a Banach Algebra . . . . . . . . . . . . . . . . . 2989.6 Banach Algebras with Involution; C∗-Algebras . . . . . . . . . . . . . . . . 3019.7 Characters and Gelfand Transform in a C∗-Algebra . . . . . . . . . . . . . . 3079.8 The Enveloping C∗-Algebra of an Involutive Banach Algebra . . . . . . . . 310

CONTENTS 9

10 Analysis on Locally Compact Abelian Groups 31310.1 Characters and The Dual Group . . . . . . . . . . . . . . . . . . . . . . . . 31810.2 The Fourier Transform and the Fourier Cotransform . . . . . . . . . . . . . 33110.3 The Fourier Transform on a Finite Abelian Group . . . . . . . . . . . . . . 33910.4 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34310.5 Fourier transform and Cotransform in Terms of Matrices . . . . . . . . . . . 34610.6 The Discrete Fourier Transform (on Z/nZ) . . . . . . . . . . . . . . . . . . 35310.7 Some Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . 35810.8 Plancherel’s Theorem and Fourier Inversion . . . . . . . . . . . . . . . . . . 36010.9 Pontrjagin Duality and Fourier Inversion . . . . . . . . . . . . . . . . . . . . 363

11 Hilbert Algebras 36911.1 Representations of Algebras with Involution . . . . . . . . . . . . . . . . . . 37211.2 Positive Linear Forms and Positive Hilbert Forms . . . . . . . . . . . . . . . 37911.3 Traces, Bitraces, Hilbert Algebras . . . . . . . . . . . . . . . . . . . . . . . 38111.4 Complete Separable Hilbert Algebras . . . . . . . . . . . . . . . . . . . . . . 38611.5 The Plancherel–Godement Theorem ~ . . . . . . . . . . . . . . . . . . . . . 398

12 Representations of Locally Compact Groups 40912.1 Finite-Dimensional Group Representations . . . . . . . . . . . . . . . . . . . 41012.2 Unitary Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . 41412.3 Unitary Representations of G and L1(G) . . . . . . . . . . . . . . . . . . . . 41912.4 Functions of Positive Type and Unitary Representations . . . . . . . . . . . 42712.5 The Gelfand–Raikov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 43312.6 Measures of Positive Type and Unitary Representations . . . . . . . . . . . 437

13 Analysis on Compact Groups and Representations 44513.1 The Peter–Weyl Theorem, I . . . . . . . . . . . . . . . . . . . . . . . . . . . 44913.2 Characters of Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . 45813.3 The Peter–Weyl Theorem, II . . . . . . . . . . . . . . . . . . . . . . . . . . 46413.4 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

14 Induced Representations 48114.1 Cocycles and Induced Representations . . . . . . . . . . . . . . . . . . . . . 48414.2 Cocycles on a Homogeneous Space X = G/H . . . . . . . . . . . . . . . . . 48814.3 Converting Induced Representations of G From EX to EG . . . . . . . . . . 49314.4 Construction of the Hilbert Space L2µ(X;E) . . . . . . . . . . . . . . . . . . 49514.5 Induced Representations, I; G/H has a G-Invariant Measure . . . . . . . . . 49814.6 Quasi-Invariant Measures on G/H . . . . . . . . . . . . . . . . . . . . . . . 50114.7 Induced Representations, II; Quasi-Invariant Measures . . . . . . . . . . . . 50414.8 Partial Traces, Induced Representations of Compact Groups . . . . . . . . . 512

15 Harmonic Analysis on Gelfand Pairs 521

10 CONTENTS

15.1 Gelfand Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52415.2 Real Forms of a Complex Semi-Simple Lie Group . . . . . . . . . . . . . . . 52915.3 Spherical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54615.4 Examples of Gelfand Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55515.5 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56815.6 The Plancherel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57115.7 Extension of the Plancherel Transform; P(G) and P′(Z) ~ . . . . . . . . . . 57815.8 Spherical Functions of Positive Type and Representations . . . . . . . . . . 583

A Topology 587A.1 Metric Spaces and Normed Vector Spaces . . . . . . . . . . . . . . . . . . . 587A.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594A.3 Continuous Functions, Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 604A.4 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611A.5 Compact Sets and Locally Compact Spaces . . . . . . . . . . . . . . . . . . 621A.6 Second-Countable and Separable Spaces . . . . . . . . . . . . . . . . . . . . 633A.7 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637A.8 Complete Metric Spaces and Compactness . . . . . . . . . . . . . . . . . . . 643A.9 Completion of a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . 646A.10 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . 654A.11 Continuous Linear and Multilinear Maps . . . . . . . . . . . . . . . . . . . . 658A.12 Completion of a Normed Vector Space . . . . . . . . . . . . . . . . . . . . . 665A.13 Futher Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

B Vector Norms and Matrix Norms 669B.1 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669B.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

C Basics of Groups and Group Actions 689C.1 Groups, Subgroups, Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 689C.2 Group Actions: Part I, Definition and Examples . . . . . . . . . . . . . . . 702C.3 Group Actions: Part II, Stabilizers and Homogeneous Spaces . . . . . . . . 715C.4 The Grassmann and Stiefel Manifolds . . . . . . . . . . . . . . . . . . . . . 723

D Hilbert Spaces 729D.1 The Projection Lemma, Duality . . . . . . . . . . . . . . . . . . . . . . . . 729D.2 Total Orthogonal Families, Fourier Coefficients . . . . . . . . . . . . . . . . 742D.3 The Hilbert Space `2(K) and the Riesz-Fischer Theorem . . . . . . . . . . . 750

Bibliography 761

Chapter 1

Introduction

The main topic of this book is the Fourier transform and Fourier series, both understood ina broad sense.

Historically, trigonometric series were first used to solve equations arising in physics, suchas the wave equation or the heat equation. D’Alembert used trigonometric series (1747) tosolve the equation of a vibrating string, elaborated by Euler a year later, and then solvedin a different way essentially using Fourier series by D. Bernoulli (1753). However it wasFourier who introduced and developed Fourier series in order to solve the heat equation, ina sequence of works on heat diffusion, starting in 1807, and culminating with his famousbook, Théorie analytique de la chaleur , published in 1822.

Originally, the theory of Fourier series is meant to deal with periodic functions on thecircle T = U(1), say functions with period 2π. Remarkably the theory of Fourier series iscaptured by the following two equations:

f(θ) =∑m∈Z

cmeimθ. (1)

cm =

∫ π−πf(θ)e−imθ

dθ

2π. (2)

Equation (1) involves a series, and Equation (2) involves an integral. There are two waysof interpreting these equations.

The first way consists of starting with a convergent series as given by the right-hand sideof (1) (of course cn ∈ C), and to ask what kind of function is obtained. A second question isthe following: Are the coefficients in (1) computable in terms of the formulae given by (2)?

The second way is to start with a periodic function f , apply Equation (2) to obtain the cm,called Fourier coefficients , and then to consider Equation (1). Does the series

∑m∈Z cme

imθ

(called Fourier series) converge at all? Does it converge to f?

Observe that the expression f(θ) =∑

m∈Z cmeimθ may be interpreted as a countably infi-

nite superposition of elementary periodic functions (the harmonics), intuitively representing

11

12 CHAPTER 1. INTRODUCTION

simple wave functions, the functions θ 7→ eimθ. We can think of m as the frequency of thiswave function.

The above questions were first considered by Fourier. Fourier boldly claimed that everyfunction can be represented by a Fourier series. Of course, this is false, and for serevalreasons. First, one needs to define what is an integrable function. Second, it depends onthe kind of convergence that are we dealing with. Remarkably, Fourier was almost right,because for every function f in L2(T), a famous and deep theorem of Carleson states thatits Fourier series converges to f almost everywhere in the L2-norm.

Given a periodic function f , the problem of determining when f can be reconstructed asthe Fourier series (Equation (1)) given by its Fourier coefficients cm (Equation (2)) is calledthe problem of Fourier inversion. To discuss this problem, it is useful to adopt a moregeneral point of view of the correspondence between functions and Fourier coefficients, andFourier coefficients and Fourier series.

Given a function f ∈ L1(T), Equation (2) yields the Z-indexed sequence (cm)m∈Z ofFourier coefficients of f , with

cm =

∫ π−πf(θ)e−imθ

dθ

2π,

which we call the Fourier transform of f , and denote by f̂ , or F(f). We can view the Fouriertransform F(f) of f as a function F(f) : Z→ C with domain Z.

On the other hand, given a Z-indexed sequence c = (cm)m∈Z of complex numbers cm, wecan define the Fourier series F(c) associated with c, or Fourier cotransform of c, given by

F(c)(θ) =∑m∈Z

cmeimθ.

This time, F(c) is a function F(c) : T→ C with domain T. Fourier inversion can be statedas the equation

f(θ) = ((F ◦ F)(f))(θ).

Of course, there is an issue of convergence. Namely, in general, f̂ = F(f) does notbelong to L1(T). There are special cases for which Fourier inversion holds, in particular, iff ∈ L2(T).

Let us now consider the Fourier transform of (not necessarily periodic) functions defined

on R. For any function f ∈ L1(R), the Fourier transform f̂ = F(f) of f is the functionF(f) : R→ C defined on R given by

f̂(x) = F(f)(x) =∫Rf(y)e−iyx

dx(y)√2π

,

and the Fourier cotransform F(f) of f is the function F(f) : R→ C defined on R given by

Ff(x) =∫Rf(y)eiyx

dx(y)√2π

.

13

This time, the domain of the Fourier transform is the same as the domain of the Fouriercotransform, but this is an exceptional situation. Also, in general the Fourier transform f̂ isnot integrable, so Fourier inversion only holds in special cases.

The preceding examples suggest two questions:

(1) What is the input domain of the Fourier transform?

(2) What is the output domain of the Fourier transform?

The answer to (1) is that the domain of the Fourier transform, denoted by F , is a setof functions on a group G. In order for the Fourier transform to be useful, it should behavewell with respect to an operation on the set of functions on G called convolution (denotedf ∗ g), which implies that these functions should be integrable.

This leads to the first subtopic, which is: what is integration on a group? The technicalanswer involves the Haar measure on a locally compact group. Thus, any serious effortto understand what the Fourier transform is entails learning a certain amount of measuretheory and integration theory, passing through versions of the Radon–Riesz theorem relatingRadon functionals and Borel measures, and culminating with the construction of the Haarmeasure. This preliminary material is discussed in Chapters 2, 3, 4, 5, 6, and 7.

Chapter 2 gathers some basic results about function spaces, in particular, about differenttypes of convergence (pointwise, uniform, compact). Some sophisticated notions cannot beavoided, such as equicontinuity, filters, topologies defined by semi-norms, and Fréchet spaces.

Chapter 3 provides a quick review of the Riemann integral and its generalization toregulated functions.

Chapter 4 is devoted to basics of measure theory: σ-algebras, semi-algebras, measurablespaces, monotone classes, (positive) measures, measure spaces, null sets, and propertiesholding almost everywhere. We also define outer measures and state Carathéodory’s theoremwhich gives a method for constructing a measure from an outer measure. We concludeby using Carathéodory’s theorem to define the Lebesgue measure on R and Rn from theLebesgue outer measure. Our presentation relies on Halmos [39], Rudin [67], Lang [53], andSchwartz [73].

Chapter 5 develops the theory of Lebesgue integration in a fairly general context, namelyfunctions defined on a measure space taking values in a Banach space. This integral isusually known as the Bochner integral (developed independently by Dunford). We agreewith Lang (Lang [53]) that the investment needed to deal with functions taking values in aBanach space rather than in R is minor, and that the reward is worthwhile. This approachis presented in detail in Dunford and Schwartz [28], and more recent (and easier to read)expositions of this method are given in Lang [53] and Marle [57].

After reading this chapter, the reader will know what are the spaces L1(X), L2(X), andL∞(X), which is essential to move on to the study of harmonic analysis. In this chapter, weprovide some proofs.


Chapter 6 presents the theory of integration on locally compact spaces due to Radonand Riesz based on linear functionals on the space of continuous functionals with compactsupport. Although this material is well-known to analysts, it may be less familiar to othermathematicians, and most computer scientists have not been exposed to it. Our presentationrelies heavily on Rudin [67] (Chapter 2), Lang [53] (Chapter IX), Folland [32] (Chapter 7),Marle [57], and Schwartz [73]. We also borrowed much from Dieudonné [22] (Chapter XIII).

We state the famous representation theorem of Radon and Riesz for positive linear func-tionals and certain types of positive Borel measures (Theorem 6.8 and Theorem 6.10). Here,inspired by Folland and Lang, we define a σ-Radon measure as a Borel measure which isouter regular, σ-inner regular, and finite on compact subsets. A Radon measure is a σ-Radonmeasure which is also inner regular. Linear functionals which are bounded on the space ofcontinuous functions with support contained in a fixed compact support are called Radonfunctionals . We have avoided Bourbaki and Dieudonné’s use of the term Radon measure fora Radon functional, which is just too confusing.

We define complex measures, and following Rudin, we present the Radon–Riesz corre-spondence between bounded Radon functionals and complex (regular) measures (Theorem6.29). This theorem is absolutely crucial to the construction of the Haar measure and to thedefinition of the convolution of complex measures and of functions.

Chapter 7 contains a rather complete discussion of the Haar measure on a locally com-pact group, convolution, and the application of convolution to regularization. After somepreliminaries about topological groups (Section 7.1), we describe the method for construct-ing a left Haar measure from a left Haar functional, following essentially Weil’s proof aspresented in Folland [31] (see Sections 7.2 and 7.3). We prove almost everything, except fora technical lemma. Then we prove the uniqueness of the left Haar measure up to a positiveconstant, using Dieudonné’s method [22] (Section 7.4). We introduce the modular functionand the modulus of an automorphism. We show how to use the Haar measure to construct ahermitian inner product invariant under the representation of a compact group. We discussG-invariant measures on homogeneous spaces.

One of the main applications of the Haar measure is the definition of the convolutionµ ∗ ν of (complex) measures and the convolution f ∗ g of functions; see Section 7.11. Un-der convolution, the set M1(G) of complex regular measures is a Banach algebra with aninvolution, and a multiplicative unit element. This algebra contains the Banach subalgebraL1(G), which doesn’t have a multiplicative unit in general. In Section 7.12, we show that byconvolving a function f with functions gn from a “well-behaved” family we obtain a sequence(f ∗ gn) of functions more regular that f that converge to f . This technique is known asregularization.

Chapter 7 is the last of the chapters dealing with background material. Similar materialis coved in Folland [31], and very extensively in Hewitt and Ross [44] (over 400 pages).

The main chapters presenting some elements of harmonic analysis, in particular theFourier transform, are:

15

1. Chapter 8, in which the classical theory of the Fourier transform (and cotransform)on T, R, and then Tn and Rn, is presented. We also present the sampling theoremdue to Shannon, and discuss the Heisenberg uncertainty principle. Our presentation isinspired by Rudin [67], Folland [30, 32], Stein and Shakarchi [78], and Malliavin [56].

2. Chapter 10, which is devoted to harmonic analysis on locally compact abelian groups ,based on the seminal work of A. Weil, Gelfand, and Pontrjagin. Our presentation isbased on Folland [31] and Bourbaki [8].

3. Chapter 13, which gives an exposition of harmonic analysis on a compact not necessarilyabelian group G. The main result is the beautiful theorem of Peter and Weyl, whichamong other things, gives the structure of the algebra L2(G) as a Hilbert sum of finite-dimensional spaces corresponding to irreducible representations of G. We rely heavilyon Dieudonné [22, 19], Folland [31], and Hewitt and Ross [43].

4. Chapter 15, which presents a theory of the Fourier transform that generalizes all previ-ous definitions, based on the concept of a Gelfand pair (G,K). We follow Dieudonné’sexposition in [20].

Chapters 10, 13, and 15, require more preparatory material.

If G is a commutative locally compact group, then the domain of the Fourier transformon L1(G) is the group Ĝ of characters of G, the homomorphisms χ : G → C such that|χ(g)| = 1 for all g ∈ G. The group Ĝ is called the Pontrjagin dual of G. It turns outthat Ĝ is homeomorphic to the space X(L1(G)) of characters of the Banach algebra L1(G).Thus we need some knowledge about normed algebras. Chapter 9 presents the basic theoryof normed algebras and their spectral theory needed for Chapter 10. The study of algebrasand normed algebras focuses on three concepts:

(1) The notion of spectrum σ(a) of an element a of an algebra A.

(2) If A is a commutative algebra, the notion of character , and the space X(A) of charactersof A.

(3) If A is a commutative algebra, the notion of Gelfand transform, G : A→ C(X(A);C).

The Gelfand transform from L1(G) to X(L1(G)) is the Fourier cotransform on L1(G). Ourpresentation is inspired by Dieudonné [22], Bourbaki [8], and Rudin [68].

If G is a locally compact abelian group, then for any function f ∈ L1(G), the Fouriertransform F(f) of f is then a function

F(f) : Ĝ→ C.

In general, Ĝ is completely different from G, and this creates problems. For the familiarcases, G = T ∼= U(1), G = Z, G = R, and G = Z/nZ, the characters are well known. Thecase G = Z/nZ corresponds to the discrete Fourier transform.


For the groups listed above, we know that under some suitable restriction, we have Fourierinversion, which means that there is some transform F (called Fourier cotransform) suchthat

f = F(F(f)). (∗)We have to be a bit careful because the domain of F is L1(Ĝ), and not L1(G), are they areusually very different beause in general G and Ĝ are not isomorphic. Then (assuming that it

makes sense), F(F(f)) is a function with domain ̂̂G, so there seems no hope, except in veryspecial cases such as G = R, that (∗) could hold. Fortunately, Pontrjagin duality assertsthat G and

̂̂G are isomorphic, so (∗) holds (under suitable conditions) in the form

f = F(F(f)) ◦ η,

where η : G→ ̂̂G is a canonical isomorphism.If G is a commutative abelian group, there is a beautiful and well understood theory of

the Fourier transform based on results of Gelfand, Pontrjagin, and André Weil presentedin Chapter 10. In particular, even though the Fourier transform is not defined on L2(G) in

general, for any function f ∈ L1(G) ∩ L2(G), we have F(f) ∈ L2(Ĝ), and by Plancherel’stheorem, the Fourier transform extends in a unique way to an isometric isomorphism between

L2(G) and L2(Ĝ). Furthermore, if we identify G and̂̂G by Pontrjagin duality, then F and

F are mutual inverses.If G is not commutative, things are a lot tougher. Characters no longer provide a

good input domain, and instead one has to turn to unitary representations . A unitaryrepresentation is a homomorphism U : G → U(H) satisfying a certain continuity property,where U(H) is the group of unitary operators on the Hilbert space H. Then Ĝ is the set ofequivalence classes of irreducible unitary representations of G, but it is no longer a group.

Chapters 11 and 12 provide the background material needed in Chapter 13. Chapter 11discusses representations of algebras, and gives an introduction to Hilbert algebras. For ourpurposes, the most important example of a complete Hilbert algebra is L2(G), where G is acompact (metrizable) group. One of the main theorems of this chapter is a structure theoremfor complete separable algebras (Theorem 11.27). This theorem is the key result for provinga major part of the Peter–Weyl theorem in Chapter 13. We follow closely Dieudonné [22].

Chapter 12 gives a brief introduction to the theory of unitary representations of locallycompact groups. We prove that there is a bijection between unitary representations of alocally compact group G and nondegenerate representations of the algebra L1(G). We definefunctions and measures of positive type, and prove that there is a bijection between the set offunctions of positive type and cyclic unitary representations (Gelfand–Raikov, Godement).We follow Dieudonné [19, 20] and Folland [31].

One more preparatory chapter is needed for Chapter 15. Chapter 14 gives an introductionto induced representations. The goal is to construct a unitary representation of a group Gfrom a representation of a closed subgroup H of G.

17

A way to deal with noncommutativity, due to Gelfand, is to work with pairs (G,K) whereK is a compact subgroup of G. This theory is presented in Chapter 15. Then, instead ofworking with functions on G, which is “too big,” we work with functions on the homogeneousspace G/K, the space of left cosets. Under certain assumptions on G and K, which makes(G,K) a Gelfand pair , it is possible to consider a commutative algebra of functions on theset of double cosets KsK (s ∈ G), so that some results from the commutative theory can beused. The domain of the Fourier transform is a set of functions called spherical functions ,and this set happens to be homeomorphic to the set of characters on the commutative algebramentioned above. There is a very nice theory of the Fourier transform and its inverse, buthow useful it is in practice remains to be seen.

More basic background material dealing with elementary topology, matrix norms, groupsand group actions, and Hilbert spaces is found in Appendices A, B, C, and D. These chaptersshould be considered as appendices and should be consulted by need.

Even though the present document is already quite long, it is by no means complete. Ifa locally compact group is a Lie group, then the whole machinery of Lie algebras and Liegroups developed by Élie Cartan and Hermann Weyl involving weights and roots becomesavailable. In particular, if G is a connected semisimple Lie group, there is a beautiful andextensive theory of harmonic analysis due to Harish–Chandra. We lack the expertise todiscuss this difficult theory and refer the ambitious reader to Warner’s monographs [85, 86],and Helgason’s treatises [42], [41] (especially Chapter IV), and [40] (especially Chapter III,Section 12).

To keep the length of this book under control, we resigned ourselves to omit many proofs.This is unfortunate because some beautiful proofs (such as the proof of the Radon–Riesztheorem for bounded Radon functional) had to be omitted. However, whenever a proof isomitted, we provide precise pointers to sources where such a proof is given.

Chapter 2

Function Spaces Often Encountered

Various spaces of functions f : E → F from a topological space E to a metric space or anormed vector space F come up all the the time. The most frequently encountered arethe spaces (FE)b of bounded functions, the spaces K(E;F ) of continuous functions withcompact support, the spaces C0(E;F ) of continuous functions which tend to zero at infinity,and the spaces Cb(E;F ) of bounded continuous functions. When F is a normed vector space,all these spaces are normed vector spaces with the sup norm. An important issue aboutfunction spaces is the convergence of sequences of functions. We review the main threenotions, pointwise convergence (also known as simple convergence), uniform convergence,and compact convergence. A sequence of continuous functions may converge pointwise toa function which is not continuous. Uniform convergence has a better behavior. If F is acomplete normed vector space, then both spaces Cb(E;F ) and (FE)b are also complete. Aninteresting family of functions in (F [a,b])b is the space Reg([a, b];F ) of regulated functions.These functions only have at most countably many simple kinds of discontinuties calleddiscontinuities of the first kind. If F is a complete normed vector space, then the spaceStep([a, b];F ) is complete. It contains a subspace Step([a, b];F ) consisting of very simplefunctions called step functions, which take finitely many different values on consecutiveopen intervals. The space Step([a, b];F ) is dense in Reg([a, b];F ). If E is a localy compactspace, then the space C0(E;C) is the closure of KC(E) in Cb(E;C).

2.1 Spaces of Bounded Functions

In this section, we are dealing with functions f : E → F , where F is either a metric space ora normed vector space. Recall that the set of all functions from E to F is denoted by FE.

First assume that F is a metric space with metric d. We would like to make FE into ametric space. It is natural to define a metric on FE by setting

d∞(f, g) = supx∈E

d(f(x), g(x))

for any two functions f, g : E → F , but if d(f(x), g(x)) is unbounded as x ranges over E, the

19

20 CHAPTER 2. FUNCTION SPACES OFTEN ENCOUNTERED

expression supx∈E d(f(x), g(x)) is undefined. Therefore, we consider the space of boundedfunctions defined as follows.

Definition 2.1. If (F, d) is a metric space, a function f : E → F is bounded if its imagef(E) is bounded in F , which means that f(E) ⊆ B(a, α), for some closed ball B(a, α) ofcenter a and radius α > 0. See Figure 2.1. The space of bounded functions f : E → F isdenoted by (FE)b.

1 1

0

-1

i.

ii.

f

ff( )R

Figure 2.1: Let E = F = R with the Euclidean metric. In Figure (i), f is unbounded sincef(E) = R. In Figure (ii), f ∈ (FE)b since f(E) = (0, 1] and (0, 1] ⊂ B(0, 1) = [−1, 1].

If f : E → F and g : E → F are bounded functions then it is easy to see that if f(E) ⊆B(a, α) and if g(E) ⊆ B(b, β), then

d(f(x), g(x)) ≤ α + β + d(a, b) for all ∈ E.

Therefore, supx∈E d(f(x), g(x)) is well defined. It is easy to check that if we define


d(f(x), g(x))

for any two bounded functions f, g, then d is indeed a metric on (FE)b.

Definition 2.2. If (F, d) is a metric space, then for any two bounded functions f, g ∈ (FE)b,the quantity


d(f(x), g(x))

is a metric on (FE)b. See Figure 2.2.

If (F, ‖ ‖) is normed metric space, then FE is a vector space, and it is easy to checkthat (FE)b is also a vector space. For any bounded function f : E → F (which means thatf(E) ⊆ B(0, α), for some closed ball B(0, α)), then

‖f‖∞ = supx∈E‖f(x)‖

is a norm on the vector space (FE)b.

2.2. CONVERGENCE: POINTWISE, UNIFORM, COMPACT 21

1

f

-1

g

Figure 2.2: Let E = F = R with the Euclidean metric. Both f, g ∈ (FE)b since f(E) = (0, 1),while g(E) = [−1, 0). The concatenation of the vertical dashed red lines is d∞(f, g) =supx∈E d(f(x), g(x)) = 1− (−1) = 2.

Definition 2.3. If (F, ‖ ‖) is a normed vector space, then for any bounded function f ∈(FE)b, the quantity

‖f‖∞ = supx∈E‖f(x)‖

is a norm on (FE)b, often called the sup norm.

The following important theorem can be shown; see Schwartz [70] (Chapter XV, Section1, Theorem 1).

Theorem 2.1. (1) If (F, d) is a complete metric space, then ((FE)b, d∞) is also a completemetric space.

(2) If (F, ‖ ‖) is a complete normed vector space, then ((FE)b, ‖ ‖∞) is also a completenormed vector space.

2.2 Convergence: Pointwise, Uniform, Compact

When dealing with spaces of functions, a crucial issue is to identify notions of limit thatpreserve certain desirable properties, such as continuity. There are primarily three suchnotions of convergence: pointwise, uniform, and compact, that we now review.

Definition 2.4. Let (F, d) be a metric space. A sequence (fn)n≥1 of functions fn : E → Fconverges pointwise (or converges simply) to a function f : E → F if for every x ∈ E, forevery � > 0, there is some N > 0 such that

d(fn(x), f(x)) < � for all n ≥ N.

See Figure 2.3.


f

f

f

f

f

f

f

1

2

3

4

n

n+1

x

f(x)f(x) + ε

f(x) - ε

Figure 2.3: A schematic illustration of fn(x) converging pointwise f(x), where E = F = R.As n increases, the graph of fn(x) near x must be in the band determined by the graphs off(x)− � and f(x) + �.

Definition 2.4 says that for every x ∈ E, the sequence (fn(x))n≥1 converges to f(x).Observe that the above � depends on x.

Let F be a topological space. The product topology on FE is defined as follows: a subsetof functions in FE is open if it is the union of subsets UA of functions f : E → F for whichthere is some finite subset A of E such that f(x) ∈ Ux for all x ∈ A, where Ux is an opensubset of F , and f(x) ∈ F is arbitrary for all x ∈ E − A. Equivalently, for any x ∈ E andany open subset U of F , let S(x, U) be the set

S(x, U) = {f | f ∈ FE, f(x) ∈ U}.

Then observe thatUA =

⋂x∈A

S(x, Ux), A finite,

that is, the sets S(x, U) form a subbasis of the product topology on FE.

For every x ∈ E, if πx : FE → F is the projection map given by

πx(f) = f(x), f ∈ FE,

(evaluation at x), then the product topology on FE is the weakest topology that makes allthe πx continuous. Indeed, the weakest topology on F

E making all the πx continuous consistsof all unions of finite intersections of subsets of FE of the form π−1x (Ux), for any open subsetUx of F , but

π−1x (Ux) = S(x, Ux),

is one of the sets in the subbasis defined above. For this reason, the product topology onFE is also called the weak topology induced by the family of functions (πx)x∈E; see Rudin[68] (Chapter 3, Section 3.8). Then it is not hard to see that a sequence (fn)n≥1 of elements


of FE converges pointwise to f ∈ FE iff the sequence (fn)n≥1 converges to f in the producttopology; see Munkres [63] (Chapter 7, Section 46, Theorem 46.1), or Folland [32] (Chapter4, Proposition 4.2). This is sometimes referered to as weak convergence.

Definition 2.5. If F is any topological space and E is any set, the topology on FE havingthe sets

S(x, U) = {f | f ∈ FE, f(x) ∈ U}, x ∈ E, U open in F ,

as a subbasis is the topology of pointwise convergence. An open subset of FE in this topologyis any union (possibly infinite) of finite intersections of subsets of the form S(x, U) as above.

If F is Hausdorff, so is the topology of pointwise convergence. Indeed, if f, g ∈ FE andf 6= g, then there is some x ∈ E such that f(x) 6= g(x), and since F is Hausdorff, thereexist two disjoint open subsets Uf(x) and Ug(x) with f(x) ∈ Uf(x) and g(x) ∈ Ug(x). Thenπ−1x (Uf(x)) and π

−1x (Ug(x)) are disjoint open subsets with f ∈ π−1x (Uf(x)) and g ∈ π−1x (Ug(x)).

Definition 2.6. Let (F, d) be a metric space. A sequence (fn)n≥1 of functions fn : E → Fconverges uniformly to a function f : E → F if for every � > 0, there is some N > 0 suchthat

d(fn(x), f(x)) < � for all n ≥ N and for all x ∈ E.

See Figure 2.4.

f

f + ε

f - ε

f

f + ε

f - ε

f1

f2

f4

f

fn

f n+1

3

Figure 2.4: A schematic illustration of fn converging uniformly to f , where E = F = R. Asn increases, the graph of fn must lie entirely in the band determined by the graphs of f − �and f + �.


The difference between simple and uniform convergence is that in uniform convergence,� is independent of x. For example the functions fn : [0, 2π] → R defined by fn(x) =n sin

(xn

)converges uniformly to f(x) = x, as evidenced by Figure 2.5. Consequently, uniform

convergence implies simple convergence, but the converse is false, as the following examplesillustrate.

x4 2

3 4

5 4

3 2

7 4

2

K1

0

1

2

3

4

5

6

Figure 2.5: The colored functions fn(x) = n sin(xn

), over the domain [0, 2π], converge uni-

formly to the black line f(x) = x.

Example 2.1. Let g : R→ R be the function given by

g(x) =1

1 + x2,

and for every n ≥ 1, let fn : R→ R be the function given by

fn(x) =1

1 + (x− n)2.

The function fn is obtained by translating g to the right using the translation x 7→ x + n;see Figure 2.6

Since

limn7→∞

1

1 + (x− n)2= 0,

the sequence (fn)n≥1 converges pointwise to the zero function f given by f(x) = 0 for allx ∈ R. However, since the the maximum of each fn is 1, we have

d∞(fn, f) = 1 for all n ≥ 1,

so the sequence (fn)n≥1 does not converge uniformly to the zero function.

2.2. CONVERGENCE: POINTWISE, UNIFORM, COMPACT 25:

xK15 K10 K5 0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 2.6: The bell curve graphs of Example 2.1; g(x) in brown; f1(x) in red; f2(x) inpurple; f3(x) in blue.

Example 2.2. Pick any positive real α > 0. For each n ≥ 1, let fn : R→ R be the piecewiseaffine function defined as follows:

fn(x) =

0 if x ≤ 0 or x ≥ 1/n(2n)nαx if 0 ≤ x ≤ 1/(2n)2nα(1− nx) if 1/(2n) ≤ x ≤ 1/n.

See Figure 2.7.

xK2 K1 0 1 2

10

20

30

40

50

60

Figure 2.7: The piecewise affine functions of Example 2.2 with α = 3; f1(x) in magenta;f2(x) in red; f3(x) in purple; f3(x) in blue. Each fn(x) has a symmetrical triangular peak.As n increases, the peak becomes taller and thinner.

For every x > 0, there is some n such that 1/n < x, so limn7→∞ fn(x) = 0 for x > 0, andsince fn(x) = 0 for x ≤ 0, we see that the sequence (fn)n≥1 converges pointwise to the zero


function f . However, the maximum of fn is nα (for x = 1/(2n)) so

d∞(fn, f) = nα,

and limn7→∞ d∞(fn, f) =∞, so the sequence (fn)n≥1 does not converge uniformly to the zerofunction.

Observe that convergence in the metric space of bounded functions ((FE)b, d∞) is theuniform convergence of sequences of functions. Similarly, convergence in the normed vectorspace of bounded functions ((FE)b, ‖ ‖∞) is the uniform convergence of sequences of func-tions. For this reason, the topology on (FE)b induced by the metric d∞ (or the norm ‖ ‖∞)is sometimes called the topology of uniform convergence.

If E is a topological space, it is useful to define the following notion of convergence.

Definition 2.7. Let E be a topological space and let (F, d) be a metric space. A sequence(fn)n≥1 of functions fn : E → F converges locally uniformly to a function f : E → F if forevery x ∈ E, there is some open subset U of E containing x such that for every � > 0, thereis some N > 0 such that

d(fn(x), f(x)) < � for all n ≥ N and for all x ∈ U.

If E is locally compact, it is easy to see that a sequence (fn)n≥1 converges locally uniformlyiff it converges uniformly on every compact subset of E.

If a sequence (fn)n≥1 of continuous functions converges pointwise to a function f , thelimit f is not necessarily continuous. For example, the functions fn : [0, 1] → R given byfn(x) = x

n are continuous, and the sequence (fn)n≥1 converges pointwise to the discontinuousfunction f : [0, 1]→ R given by

f(x) =

{0 if 0 ≤ x < 11 if x = 1,

as evidenced by Figure 2.8.The following theorem gives sufficient conditions for the limit of a sequence of continuous

functions to be continuous.

Theorem 2.2. Let E be a topological space, and (F, d) be a metric space, and let (fn)n≥1be a sequence of functions fn : E → F converging locally uniformly to a function f : E → F .Then the following properties hold:

(1) If the functions fn are continuous at some point a ∈ E, then the limit f is alsocontinuous at a.

(2) If the functions fn are continuous (on the whole of E), then the limit f is also contin-uous (on the whole of E).


x0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 2.8: The sequence of functions fn(x) = xn over [0, 1] converges pointwise to the

discontinuous green graph.

(3) If E is metric space, the sequence (fn)n≥1 converges uniformly to f , and the fn areuniformly continuous on E, then the limit f is also uniformly continuous on E.

The proof of Theorem 2.2 can be found in Schwartz [70] (Chapter XV, Section 4, Theorem1).

Here are a few applications of Theorem 2.2.

Definition 2.8. Let E be a topological space, and let (F, d) be a metric space. The metricsubspace of ((FE)b, d∞) consisting of all continuous bounded functions f : E → F is de-noted Cb(E;F ). If (E, ‖ ‖) is a normed vector space, the normed subspace of ((FE)b, ‖ ‖∞)consisting of all continuous bounded functions f : E → F is also denoted Cb(E;F ).

Proposition 2.3. Let E be a topological space, and let (F, d) be a metric space. The met-ric subspace Cb(E;F ) of ((FE)b, d∞) is closed. If (F, d) is a complete metric space, then(Cb(E;F ), d∞) is also complete.

Proposition 2.4. Let E be a topological space, and let (F, ‖ ‖) be a normed vector space.The normed subspace Cb(E;F ) of ((FE)b, ‖ ‖∞) is closed. If (F, ‖ ‖) is a complete normedvector space, then (Cb(E;F ), ‖ ‖∞) is also complete.

An important special case of Proposition 2.4 is the case where F = R or F = C, namely,our functions are real-valued continuous and bounded functions f : E → R, or complex-valued continuous and bounded functions f : E → C. The spaces of functions (Cb(E;R), d ∞)and (Cb(E;C, ‖ ‖∞) are complete.


If E is compact and if (F, ‖ ‖) is a complete normed vector space, then every continuousfunction f : E → F is bounded. As a consequence, the space C(E;F ) of continuous functionsf : E → F is complete.

Another topology on a function space that will be used in the definition of the dual ofan abelian locally compact group is the topology of compact convergence.

Definition 2.9. Let E be any topological space and let (F, d) be a metric space. For any� > 0, any f ∈ FE, and any compact subset K of E, define the set BK(f, �) by

BK(f, �) =

{g ∈ FE | sup

x∈Kd(f(x), g(x)) < �

}.

The family of sets BK(f, �) is a subbasis of the topology of compact convergence; that is,an open set of FE in this topology is any union (possibly infinite) of finite intersections ofsubsets of the form BK(f, �).

The difference between this topology and the topology of pointwise convergence is thata general basis subset containing a function f contains functions that are close to f notjust at finitely many points, but at all points of some compact subset. Thus the topologyof pointwise convergence is weaker than the topology of compact convergence, which itselfis weaker than the topology of uniform convergence. It is easy to see the sets BK(f, �)actually form a basis of the topology of compact convergence (they are closed under finiteintersections).

It is easy to show that a sequence (fn) of functions in FE converges to a function f in

the topology of compact convergence iff for every compact subset K of E, the sequence (fn)converges uniformly to f on K.

If the space E is compactly generated, then the topology of compact convergence is evenbetter behaved.

Definition 2.10. A topological space E is compactly generated if any subset U of E is openif and only if U ∩K is open in K for every compact subset K.

The following result is shown in Munkres [63] (Chapter 7, Section 46, Lemma 46.3).

Proposition 2.5. If a topological space E is locally compact or first countable, then it iscompactly generated.

A nice consequence of E being compactly generated is that, as in the case of uniformconvergence, the limit of a sequence of continuous functions that converges to a function fin the topology of compact convergence is continuous.

Proposition 2.6. Let E be a compactly generated topological space and let (F, d) be a metricspace. Then the space C(E;F ) of continuous functions from E to F is closed in FE in thetopology of compact convergence.

2.3. EQUICONTINUOUS SETS OF CONTINUOUS FUNCTIONS 29

Proposition 2.6 is proven in Munkres [63] (Chapter 7, Section 46, Theorem 46.5).

In many applications we are interested in the space C(E;F ) of continuous functions fromE to F , and in this case there is another way to define the topology of compact convergence.

Definition 2.11. Let E and F be two topological spaces. For any compact subset K of Eand any open subset U of F , let S(K,U) be the set of continuous functions

S(K,U) = {f | f ∈ C(E;F ), f(K) ⊆ U}.

The sets S(K,U) form a subbasis for a topology on C(E;F ) called the compact-open topology .An open set in the topology is any union (possibly infinite) of finite intersections of subsetsof the form S(K,U).

It is immediately verified that if F is Hausforff, then the compact-open topology onC(E;F ) is Hausdorff.

Remark: Observe that the open subsets S(x, U) of the topology of pointwise convergencecan be viewed as the result of restricting K to be a single point but relaxing f to belong toFE.

The compact-open topology is interesting in its own right and coincides with the topologyof compact convergence when F is a metric space. The following result is proven in Munkres[63] (Chapter 7, Section 46, Theorem 46.8).

Proposition 2.7. If E is a topological space and if (F, d) is a metric space, then on the spaceC(E;F ) of continuous functions from E to F the compact-open topology and the topology ofcompact convergence coincide.

A subspace of (FR)b, where F is a Banach space (a complete normed vector space), thatplays an important role in the theory of the Riemann integral, is the space of regulatedfunctions.

Another notion that is often useful to show that a sequence of continuous functions con-verges pointwise to a continuous function is the notion of an equicontinuous set of functions.

2.3 Equicontinuous Sets of Continuous Functions

Intuitively speaking equicontinuity is of sort of uniform continuity for sets of functions.

Definition 2.12. Let E be a topological space and let (F, dF ) be a metric space. A subsetS ⊆ C(E;F ) of the set of continuous functions from E to F is equicontinuous at a pointx0 ∈ E if for every � > 0, there is some open subset U ⊆ E containing x0 such thatdF (f(x), f(x0)) ≤ � for all x ∈ U and for all f ∈ S. If E is also a metric space with metricdE, then, the above condition says that for every � > 0, there is some η > 0, such thatdF (f(x), f(x0)) ≤ � for all x ∈ E such that dE(x, x0) ≤ η, and for all f ∈ S. The set offunctions S is equicontinuous if it is equicontinuous at every point x ∈ E.


For example, if E is a metric space and if there exists two constant c, α > 0 such that wehave the Lipschitz condition

dF (f(x)− f(y)) ≤ c(dE(x, y))α, for all f ∈ S and all x, y ∈ E,

then S is equicontinuous.

Proposition 2.8. Let (fn) be a sequence of functions fn ∈ C(E;F ), and let (xn) be asequence of points xn ∈ E. If the set {fn} is equicontinuous, the sequence (xn) convergesto x ∈ E, and the sequence (fn) converges pointwise to some function f : E → F , then thesequence (fn(xn)) converges to f(x) ∈ F .

Proof. We have the inequality

dF (fn(xn), f(x)) ≤ dF (fn(xn), fn(x)) + dF (fn(x), f(x)).

For every � > 0, since the sequence (fn) converges pointwise to f , there is some N2 > 0 suchthat dF (fn(x), f(x)) ≤ �/2 for all n ≥ N2. Since {fn} is equicontinuous, there is some opensubset U ⊆ E containing x such that

dF (fn(y), fn(x)) ≤ �/2 for all n ≥ 1 and all y ∈ U.

Since (xn) converges to x, there is some N1 > 0 such that xn ∈ U for all n ≥ N1, so

dF (fn(xn), fn(x)) ≤ �/2 for all n ≥ N1,

and for all n ≥ max{N1, N2}, we have dF (fn(xn), f(x)) ≤ �, which proves that (fn(xn))converges to f(x).

There are various results about equicontinuous sets of functions usually known as variantsof Ascoli’s theorem. Schwartz [70] (Chapter XX) gives one of the most complete expositionswe are aware of. We only consider three variants of Ascoli’s theorem that we will need.

Theorem 2.9. (Ascoli I) Let E be a topological space, let (F, dF ) be a metric space, and letS ⊆ C(E;F ) be set of equicontinuous functions at some x0 ∈ E. Then the closure S of S inFE with the topology of pointwise convergence is also equicontinuous at x0. As a corollary,if S ⊆ C(E;F ) is a set of equicontinuous functions, then every function f ∈ S is continuous,and for every sequence (fn) of functions fn ∈ S, if (fn) converges pointwise to a functionf ∈ FE, then f is continuous.

Proof. Since S is equicontinuous at x0, for every � > 0, there is some open subset U containingx0 such that

dF (f(x0), f(x)) ≤ �, for all f ∈ S and all x ∈ U.But for x ∈ U fixed, the map f 7→ (f(x0), f(x)) from FE to F 2 = F × F is continuous (thisis a projection onto a product), and dF is continuous on F

2. As a consequence, the set

{f ∈ FE | dF (f(x0), f(x)) ≤ �}

2.3. EQUICONTINUOUS SETS OF CONTINUOUS FUNCTIONS 31

is closed in FE, and since it contains S, it also contains S. Thus, for every � > 0, we foundan open subset U containing x0 such that dF (f(x0), f(x)) ≤ � for all x ∈ U and all f ∈ S,which means that S is equicontinuous.

Since every function in an equicontinuous set of functions is continuous, every functionf ∈ S is continuous. By definition of the pointwise topology, if a sequence (fn) of functionsfn ∈ S converges pointwise to a function f ∈ FE, then f ∈ S, so f is continuous.

Dieudonné proves a weaker version of Theorem 2.9, namely that for every subset S ofthe space of bounded continuous functions Cb(E;F ), if S is equicontinuous, then its closureS is also equicontinuous. This is Proposition 7.5.4 in [23] (Chapter 7, Section 5).

The second version of Ascoli’s theorem involves a dense subset E0 of E. We need thefollowing variant of Definition 2.5. The topology of pointwise convergence in E0 is thetopology on FE having the sets

S(x, U) = {f | f ∈ FE, f(x) ∈ U}, x ∈ E0, U open in F ,

as a subbasis.

Theorem 2.10. (Ascoli II) Let E be a topological space, let (F, dF ) be a metric space, E0 bea dense subset of E, and S ⊆ C(E;F ) be set of equicontinuous functions. Then the topologyof pointwise convergence in E0, the topology of pointwise convergence, and the topology ofcompact convergence (all three topologies being defined in FE), induce identical topologies onS.

Theorem 2.10 is proven in Schwartz [70] (Chapter XX, Theorem XX.3.1). The followingcorollaries of Theorem 2.10 are particularly useful. The first of these two propositions is animmediate consequence of Theorem 2.10.

Proposition 2.11. Let E be a topological space and let (F, dF ) be a metric space. If asequence (fn) of continuous functions fn ∈ C(E;F ) converges pointwise to a function f ∈ FEand if {fn} is equicontinuous, then f is continuous and the sequence (fn) converges uniformlyto f on every compact subset.

Proposition 2.12. Let E be a topological space, E0 a dense subset of E, and let (F, dF ) bea metric space. If the following properties hold:

(1) The sequence (fn) of continuous functions fn ∈ C(E;F ) converges pointwise for everyx ∈ E0;

(2) The set {fn} is equicontinuous;

(3) The set {fn(x) | n ≥ 1} is contained in a complete subset of F for every x ∈ E;

then the sequence (fn) converges pointwise (for all x ∈ E) to a continuous function f , and(fn) converges uniformly to f on every compact subset. If F complete, then Condition (3)is automatically satisfied and can be omitted.


Proof. Since by Theorem 2.10, the topology of pointwise convergence on E0 is identical tothe topology of pointwise convergence on E, as the sequence (fn) converges pointwise forevery x ∈ E0, it also converges pointwise for every x ∈ E. This implies that for every x,the sequence (fn(x)) is a Cauchy sequence in F , but since by (3) the set {fn(x) | n ≥ 1} iscontained in a complete subset of F , the sequence (fn(x)) converges. Thus (fn) convergespointwise to a function f ∈ FE, and since {fn} is equicontinuous, by Proposition 2.11, thefunction f is continuous, and (fn) converges uniformly to f on every compact subset.

Dieudonné proves a special case of Proposition 2.12 where E is a metric space, F isa complete normed vector space (a Banach space), the functions fn are continuous andbounded, and {fn} is equicontinuous; see Proposition 7.5.5 and Proposition 7.5.6 in [23](Chapter 7, Section 5).

In most applications, E is a metric space and F is a (complete) normed vector space.The following result about sets of continuous linear maps will be needed.

Proposition 2.13. Let E be a metrizable vector space and F be a normed vector space. Asubset of continuous linear maps S ⊆ L(E;F ) is equicontinuous if and only if there is someopen subset V ⊆ E containing 0 and some real c > 0 such that ‖f(x)‖ ≤ c for all x ∈ V andall f ∈ S.

Proof. If S is equicontinuous then obviously the property of the proposition holds. Con-versely, for any � > 0, the condition ‖f(x)‖ ≤ c for all x ∈ V and all f ∈ S implies that‖f(x)‖ ≤ � for all x ∈ (�/c)V and all f ∈ S, so S is equicontinuous at 0. For any x0 ∈ E,and for all x ∈ x0 + (�/c)V , we have

‖f(x)− f(x0)‖ = ‖f(x− x0)‖ ≤ �

for all f ∈ S, that is, S is equicontinuous at x0.

A third version of Ascoli’s theorem involving relative compactness will be needed inSection 15.1. Recall that a subset A of a Hausforff space X is relatively compact if itsclosure A is compact in X.

Theorem 2.14. (Ascoli III) Let E be a topological space, let (F, dF ) be a metric space, andlet S ⊆ C(E;F ) be set of continuous functions. Assume the following two conditions hold:

(1) The set S is equicontinuous.

(2) For every x ∈ E, the set S(x) = {f(x) | f ∈ S} is relatively compact in F .

Then the set S is relatively compact in the space (FE)c of continuous functions from E to Fwith the topology of compact convergence. Conversely, if E is locally compact and if the setS is relatively compact in the space (FE)c, then Conditions (1) and (2) hold.

2.4. REGULATED FUNCTIONS 33

Proof. A complete proof is given in Schwartz [70] (Chapter XX, Theorem XX.4.1). Weonly prove the first part of the theorem. The proof uses Tychonoff’s powerful theorem. Byhypothesis, for every x ∈ E, the closure S(x) of S(x) is compact in F , so by Tychonoff’stheorem, the product

∏x∈E S(x) is compact in F . By definition of the above product, this

means that the set Ŝ of functions f ∈ FE such that f(x) ∈ S(x) for all x ∈ E is compact inFE with the topology of pointwise convergence. Since S is contained in the compact set Ŝ,we deduce that its closure S is compact in FE (with the topology of pointwise convergence).By Ascoli I (Theorem 2.9), since S is equicontinuous, the set S is also equicontinuous. ByAscoli II (Theorem 2.10), since the restriction to S of the topology of pointwise convergenceon FE coincides with the restriction to S of the topology of compact convergence on FE,the set S is compact in (FE)c, and thus S is relatively compact in (F

E)c.

The special case of Theorem 2.14 in which E is compact and F is a Banach space isproven in Dieudonné [23] (Chapter 7, Section 5, Theorem 5.7.5). Because F is complete theproof is simpler and does not use Tychonoff’s theorem.

2.4 Regulated Functions

Recall that there are four kinds of intervals of R: (a, b), [a, b), (a, b], and [a, b], with a < b.By convention, (a, b) = [a, b) if a =∞, and (a, b) = (a, b] if b =∞.

Definition 2.13. Let I be an interval of R, and let F be a metric space (or a normed vectorspace). Given a function f : I → F , for any x ∈ I with x 6= b, we say that f has a limit to theright in x if limy∈I, y>x f(y) exists as y ∈ I tends to x from above. This limit is denoted byf(x+). For any x ∈ I with x 6= a, we say that f has a limit to the left in x if limy∈I, y


x x x1 2 3

y

y

y

y

y

y

1

2

3

45

6

Figure 2.9: An illustration of a regulated function f : R → R. This function has threediscontinuities x1, x2, and x3, each of the first kind. Note that f(x1−) = y4, f(x1+) =f(x1) = y2, f(x2−) = f(x2) = y3, f(x2+) = y6, f(x3−) = y3, f(x3+) = y5, yet f(x3) = y1.

The function f : R→ R defined by

f(x) =

{sin(

1x

)if x 6= 0

0 if x = 0

is discontinuous at x = 0, but this is not a discontinuity of the first kind. See Figure 2.10.

x

K2 K

3 2

KK

20

23 2

2

K1

K0.5

0.5

1

Figure 2.10: The graph of f(x) = sin(

1x

), x 6= 0.

The following result is shown in Schwartz [72] (Chapter III, Section 2, Theorem 3.2.3).

Proposition 2.15. If f : I → F is a regulated function (where F is a metric space), then fhas at most countably many discontinuities of the first kind.


Regulated functions on a closed and bounded interval [a, b] must be bounded. As aconsequence, they arise as limits of uniformly convergent sequences of step functions.

Definition 2.15. A function f : R → F (where F is any set) is a step function if thereis a finite sequence (a0, a1, . . . , an) of reals such that ak < ak+1 for k = 0, . . . , n, and f isconstant on each of the open intervals (−∞, a0), (ak, ak+1) for k = 0, . . . , n, and (an,+∞).The sequence (a0, a1, . . . , an) is called an admissible subdivision for f . See Figure 2.11.

a a a aa0 1 2 3 4

-2

-1

1

2

3

Figure 2.11: An illustration of a step f : R→ R with admissible subdivision (a0, a1, a2, a3, a4).

Observe that Definition 2.15 does not make any restriction on the values f(ak), but astep function is regulated. Also, a given step function admit infinitely many admissiblesubdivisions, by refining a given subdivision. By a step function f : [a, b] → F , we mean astep function such that f(x) = 0 for all x ≤ a and for all x ≥ b.

The following result is easy to prove.

Proposition 2.16. If F is a vector space, then the set of step functions f : R → F is avector space denoted by Step(R;F ). The set of step functions f : [a, b] → F is also vectorspace denoted by Step([a, b];F ).

The following proposition is much more interesting.

Proposition 2.17. Let F be a metric space and let [a, b] be a closed and bounded interval.Then every regulated function f : [a, b] → F is the limit of a uniformly convergent sequence(fn)n≥1 of step functions fn : [a, b] → F . Furthermore, if F is a complete metric space,then the limit of any uniformly convergent sequence (fn)n≥1 of step functions is a regulatedfunction.


The proof of Proposition 2.17 is given in Schwartz [72] (Chapter III, Section 2, Theorem3.2.9).

As a corollary of Proposition 2.17, if F is a complete metric space, then the space ofregulated functions on [a, b] is closed in (F [a,b])b, and the space of step functions on [a, b] isdense in the space of regulated functions on [a, b]. Thus if F is complete, since (F [a,b])b iscomplete, the space of regulated function on [a, b] is also complete.

Another corollary of Proposition 2.17 is that every continuous function f : [a, b]→ F toa metric space F is the limit of a uniformly convergent sequence (fn)n≥1 of step functionsfn : [a, b]→ F .

If F is a vector space, the set of regulated functions defined on the closed and boundedinterval [a, b] is a vector space denoted by Reg([a, b];F ). Then, Proposition 2.17 implies thefollowing result.

Proposition 2.18. Let F be a complete normed vector space. The space Reg([a, b];F ) ofregulated functions on [a, b] is complete, and the space Step([a, b];F ) is dense in Reg([a, b];F ).

Step functions can be used to define the Riemann integral. To do so it is convenient toconsider functions of finite support.

Definition 2.16. Given any function f : E → F , where E is a topological space and F is avector space, the support supp(f) of f is the closure of the subset of E where f is nonzero,that is, supp(f) = {x ∈ E | f(x) 6= 0}. The function f has compact support if its supportsupp(f) is compact. If E is Hausdorff, this is equivalent to saying that f vanishes outsidesome compact subset K of E. See Figure 2.12.

If a step function f has compact support, then we assume that f vanishes on (−∞, a0)and on (an,+∞) for any admissible subdivision (a0, a1, . . . , an) for f .

It is easy to see that the set of continuous functions f : E → F with compact support isa vector space.

Definition 2.17. The vector space of continuous functions f : E → F with compact supportis denoted by Cc(E;F ), or K(E;F ). For every compact subset K of E, we denote by K(K;F )the space of continuous functions whose support is contained in K. Then

K(E;F ) =⋃

K⊆E, K compact

K(K;F ).

Observe that every function in K(E;F ) is bounded, that is, K(E;F ) ⊆ Cb(E;F ).If F = R or F = C, then we write KR(E) or KC(E) for K(E;F ). Radon functionals are

certain kinds of linear forms on KC(C).If (F, ‖ ‖) is a Banach space and K is a fixed compact subset of E, then so is K(K;F )

(for the sup norm ‖ ‖∞), because it is closed in Cb(E;F ). However, the normed vector space(K(E;F ), ‖ ‖∞) is not complete!


>

Figure 2.12: The graph of f : R2 → R with compact support supp = B(0, 2) = {(x, y) ∈ R2 |x2 + y2 ≤ 2}.

Example 2.3. For every n ≥ 1, consider the function un : R→ R defined as follows:

un(x) =

1 if −n ≤ x ≤ nx+ n+ 1 if −(n+ 1) ≤ x ≤ −n−x+ n+ 1 if n ≤ x ≤ n+ 10 if |x| ≥ n+ 1.

Now consider the sequence of functions (fn) given by

fn(x) = une−|x|.

Each function fn is continuous and has compact support [−(n+ 1), n+ 1], and it is easy toshow that the sequence (fn) converges uniformly to the function f given by f(x) = e

−|x|,but f does not have compact support. The problem is that the domains of the functions fn,although compact, keep growing as n goes to infinity. See Figure 2.13.

Example 2.3 shows that the normed vector space (K(E;F ), ‖ ‖∞) is not closed in thecomplete normed vector space (Cb(E;F ), ‖ ‖∞). It would be useful to identify the closureK(E;F ) of K(E;F ) in Cb(E;F ), and this can indeed be done when E is locally compact.

Assume that f belongs to the closure K(E;F ) of K(E;F ). This means that there is asequence (fn) of functions fn ∈ K(E;F ) such that limn7→∞ ‖f − fn‖∞ = 0, so for every � > 0,there is some n ≥ 1 such that ‖f(x)− fn(x)‖ ≤ � for all x ∈ E, and since fn has compact


xK6 K4 K2 0 2 4 6

0.2

0.4

0.6

0.8

xK10 K5 0 5 10

0.2

0.4

0.6

0.8

1

i.

ii.

Figure 2.13: The functions of Example 2.3. Figure (i) illustrates the u1(x) in magenta; u2(x)in red, u3(x) in orange, u4(x) in purple, and u5(x) in blue. Figure (ii) uses the same colorscheme to illustrate the corresponding fn(x). Note these fn(x) converge uniformly to greenf(x) = e−|x|.

support, there is some compact subset K of E such that ‖f(x)‖ ≤ � for all x ∈ E−K. Thissuggests the following definition.

Definition 2.18. The subspace of Cb(E;F ), denoted C0(E;F ), consisting of the continuousfunctions f such that for every � > 0, there is some compact subset K of E such that‖f(x)‖ ≤ � for all x ∈ E −K, is called the space of continuous functions which tend to 0 atinfinity .

Observe that if X = R, then a function f ∈ C0(R;F ) does indeed have the property thatlimx 7→−∞ f(x) = limx 7→+∞ f(x) = 0.

We showed that K(E;F ) ⊆ C0(E;F ). If E is locally compact, then we have the followingresult from Dieudonné [22] (Chapter XIII, Section 20) and Rudin [67] (Chapter 3, Theorem3.17).

Proposition 2.19. If E is locally compact, then C0(E;C) is the closure of K(E;C) inCb(E;C). Consequently, C0(E;C) is complete.

Proof. We already showed just before definition 2.18 that if a function f belongs to theclosure of K(E;C), then it tends to zero at infinity. Conversely, pick any f in C0(E;C).

2.5. NEIGHBORHOOD BASES AND FILTERS 39

For every � > 0, there is a compact subset K of E such that |f(x)| < � outside of K. ByProposition A.39, there is continuous function g : E → [0, 1] with compact support such thatg(x) = 1 for all x ∈ K. Clearly fg ∈ KC(E), and ‖fg − f‖∞ < �. This shows that K(E;C)is dense in C0(E;C).

In summary, if E is locally compact, then we have the strict inclusions

K(E;C) ⊂ C0(E;C) ⊂ Cb(E;C),

with C0(E;C) and Cb(E;C) complete, and K(E;C) dense in C0(E;C). It turns out that thespace of continuous linear forms on C0(E;C) is isomorphic to the space of bounded Radonfunctionals.

A modified version of step functions involving a measure will be used to define the integralon a measure space.

2.5 Neighborhood Bases and Filters

When dealing with convolution we will need a notion of convergence more general than thenotion of convergence of a sequence. There are two equivalent definitions of such a generalnotion of convergence. One in terms of nets, and the other in terms of filters. For ourpurposes, the definition in terms of filters is more convenient.

First let us review the notion of neighborhood and neighorhood base.

Definition 2.19. Let X be a topological space whose topology is specified by a set O ofopen sets. For any subset A ⊆ X, a neighborhood of A is any subset N containing some opensubset U containing A; in short, there is some U ∈ O such that A ⊆ U ⊆ N . If A = {x}, aneighborhood of {x} is called simply a neighborhood of x.

A neighborhood base of a point x (resp. of a subset A) is a family N of neighborhoods ofx (resp. of neighborhoods of A), such for every neighborhood V of x (resp. neighborhood ofA), there is some N ∈ N such that N ⊆ V .

In many cases, a neighborhood base consists of open sets. Let us now define the notionof filter and filter base. This notion is defined for any set, not just for a topological space.

Definition 2.20. Let X be any set. A filter F on X is a family of subsets of X satisfyingthe following properties.

(1) For any two subset A,B of X, if A ∈ F and if A ⊆ B, then B ∈ F (F is upward-closed).

(2) For any two subsets A,B of X, if A ∈ F and B ∈ F , then A ∩ B ∈ F (closure underintersection).

(3) We have X ∈ F .


(4) The empty set does not belong to F .

The axioms of a filter show that filters only exist on nonempty sets. In particular, Axiom(4) prevents F = 2X to be a filter.

Example 2.4.

1. If X 6= ∅, for any nonempty subset A of X, the family of all subsets of X containingA is a filter.

2. If(X,O) is a topological space, then for any x ∈ X (resp. any nonempty subset A ofX), the family of neighborhoods of x (resp. A) is a filter.

3. If X is an infinite set, the family of complements of finite subsets of X is a filter. IfX = N, then the filter of complements of finite subsets of N is called the Fréchet filter .

4. Let F be a filter on X. For any A ∈ F , let S(A) be the family

S(A) = {B ∈ F | B ⊆ A},

called a section. It is easy to check that the family of sections S(A) (for all A ∈ F) isa filter on the set F , called the filter of sections of F .

Filters are compared as follows.

Definition 2.21. Let X be any nonempty set. Given two filters F and F ′ on X, we saythat F ′ is finer than F if F ⊆ F ′.

A convenient way to generate a filter is to use a filter base.

Definition 2.22. Let X be any nonempty set. A filter base B on X is a family of subsetsof X satisfying the following properties.

(1) For any two subsets A,B of X, if A ∈ B and B ∈ B, then there is some C ∈ B suchthat C ⊆ A ∩B.

(2) The family B is nonempty.

(3) The empty set does not belong to B.

It is immediatelly verified that if B is a filter base on X, then the family of subsets of Xcontaining some subset in B is a filter called the filter generated by B.

If (X,O) is a topological space, for any x ∈ X, the filter bases of neighborhoods of x areexactly the neighborhood bases of x.

The main reason for introducing filters is to define the following general notion of con-vergence.

2.5. NEIGHBORHOOD BASES AND FILTERS 41

Definition 2.23. Let X be a topological space whose topology is specified by a set O ofopen sets. For any x ∈ X, a filter F converges to x, or x is a limit of the filter F , if everyneighborhood N of x belongs to F ; equivalently, the filter B(x) of neighborhoods of x isa subset of the filter F ; that, is the filter F is finer than the filter B(x). A filter base Bconverges to x if the filter generated by B converges to x.

The following proposition is an immediate consequence of the definition.

Proposition 2.20. Let X be a topological space whose topology is specified by a set O ofopen sets. For any x ∈ X, a filter base B converges to x iff every neighborhood base of xcontains some set in B.

Intuitively, x is a limit of a filter base B if there are sets in B as close to x as desired.The limit of a sequence (xn)n≥0 of points xn ∈ X is a special case of Definition 2.23.

Indeed, if we define for every n ≥ 0 the set Sn given by

Sn = {xp | p ≥ n},

then the family of sets Sn forms a filter base, and an element y ∈ X is a limit of the sequence(xn) iff the filter base {Sn} converges to y. Indeed, by Proposition 2.20, the filter base {Sn}converges to y iff for every neighborhood V of y, there is some Sn ⊆ V , in other words, thereis some n ≥ 1 such that xp ∈ V for all p ≥ n, which is the standard definition of convergenceof a sequence.

We can also define the notion of limit of a function. Let f : X → Y be a function whereX is any nonempty set and Y is a topological space. Then if F is any filter on X, it isimmediately verified that the family of sets f(U), with U ∈ F , is a filter base on Y .

Definition 2.24. Let f : X → Y be a function where X is any nonempty set and Y is atopological space. For any filter F on X, and for any y ∈ Y , we say that y is a limit of faccording to F (or simply that y is a limit of F) if the filter basis f(F) (consisting of thesubsets f(U) of Y with U ∈ F) converges to y. We write

limx,F

f(x) = y.

If we view a sequence (xn)n≥0 of points in a topological space X as a function x : N→ X,then (xn) converges to y in the traditional sense iff x converges to y according to the Fréchetfilter on N (the family of subsets of N having a finite complement).

The following useful characterization of a limit of a filter is immediate from the definitions.

Proposition 2.21. Let f : X → Y be a function where X is any nonempty set and Yis a topological space. A point y ∈ Y is a limit of a filter F on X if and only if for everyneighborhood V of y, there is some W ∈ F such that f(W ) ⊆ V , or equivalently, f−1(V ) ∈ Ffor every neighborhood V of y.


Filters also provide a useful characterization of the notion of compactness. First we defineultrafilters.

Definition 2.25. Let X be any nonempty set. A filter F on X is an ultrafilter if it is amaximal filter; that is, there is no filter different from F and finer than F .

For example, for any x ∈ X, the filter of subsets containing x is an ultrafilter. The follow-ing important result shows that there many ultrafilters, but they are very nonconstructivein nature. The proof uses Zorn’s lemma.

Theorem 2.22. Let X be any nonempty set. Every filter F on X is contained in a finerultrafilter.

Observe that an ultrafilter F has the following completeness property: for any subset Aof X, either A ∈ F or its complement X − A ∈ F , but not both.

Indeed, if A /∈ F and X − A /∈ F , then it is easy to see that the family G of subsets ofX given by

G = {B ⊆ X | A ∪B ∈ F}is a filter finer than F and containing X − A, thus strictly finer than F , contradicting themaximality of F . This property of ultrafilters is used in logic to prove completeness resuts.

We also need to define a notion weaker than the notion of limit of a filter.

Definition 2.26. Let X be a topological space whose topology is specified by a set O of opensets. A point x ∈ X is a cluster point (or cluster) of a filter base B if every neighborhood ofx has a nonempty intersection with every set in B (equivalently, if x ∈

⋂V ∈B V ).

A limit x of a filter is a cluster point, but the converse is false in general.

We see immediately that x is a cluster point of a filter F iff there is a filter G finer thanF and the filter G converges to x. An ultrafilter F converges to a limit x iff x is a clutserpoint of F .

Finally, we have the following characterizations of compacteness.

Theorem 2.23. Let X be a topological space whose topology is specified by a set O of opensets. The following properties are equivalent.

(1) Every filter F on X has some clutser point.

(2) Every ultafilter F on X converges to some limit.

(3) Every open cover (Uα)α∈I of X contains some finite subcover; that is, if⋃α∈I Uα = X,

then there is a finite subset J of I such that⋃α∈J Uα = X.

(4) For every family (Fα)α∈I of closed subsets of X, if⋂α∈I Fα = ∅, then there is a finite

subset J of I such that⋂α∈J Fα = ∅.

2.6. TOPOLOGIES DEFINED BY SEMI-NORMS; FRÉCHET SPACES 43

Let us also mention that a topological space X is Hausdorff if and only if every filter hasat most one limit.

The theory of filters and their use in topology is discussed quite extensively in Bourbaki[11] (Chapter 1).

2.6 Topologies Defined by Semi-Norms;

Fréchet Spaces

Certain function spaces, such as the space C(X;C) of continuous functions on a topologicalspace X, do not come with “natural” topologies defined by a norm or a metric for which theyare complete. However, the weaker notion of semi-norm can be used to define a topology,and under certain conditions, although such topologies are not defined by any norm, theyare metrizable, and complete. In this section, we briefly discuss the use of semi-norms todefine topologies. It turns out that the corresponding spaces are locally convex.

Recall from Definition B.1 that a semi-norm satisfies Properties (N2) and (N3) of anorm, but in general does not satisfy Condition (N1), so ‖x‖ = 0 does not necessarly implythat x = 0. Here is a method for defining a topology on a vector space using a family ofsemi-norms.

Definition 2.27. Let X be a vector space and let (pα)α∈I be a family of semi-norms on X.For every x ∈ X, every � > 0, and every α ∈ I, let

Ux,α,� = {y ∈ X | pα(y − x) < �}.

The topology induced by the family of semi-norms (pα)α∈I is the weakest (coarsest) topologywhose open sets are arbitrary unions of finite intersections of subsets of the form Ux,α,�.

We can think of the subset Ux,α,� as an open ball of center x and radius � in X, determinedby the semi-norm pα. We have made our vector space X into a topological space but it isnot clear that the operations (addition and scalar multiplication) are continuous. Also, ingeneral, this topology is not Hausdorff. The following proposition addresses these issues.

Proposition 2.24. Let X be a vector space and let (pα)α∈I be a family of semi-norms onX.

(1) With the topology induced by the family of semi-norms (pα)α∈I , addition and scalarmultiplication are continuous, so X is a topological vector space.

(2) For every x ∈ X, the finite intersections of subsets of the form Ux,α,� is a neighborhoodbase of x.

(3) Every open set Ux,α,� is convex.


(4) Every pα is continuous.

(5) The topology induced by the family of semi-norms is Hausdorff if and only if, for everyx 6= 0, there is some α ∈ I such that pα(x) 6= 0.

Proposition 2.24 is proven in Folland [32] (Chapter 5, Section 5.4, Theorem 5.14), orRudin [68] (Chapter 1, Theorem 1.37). In view of Property (3), the topological space X issaid to be locally convex .

Two good examples of topologies induced by families of seminorms are the topology ofpointwise convergence and the topology of comp

Aspects of Harmonic Analysis and Representation Theoryjean/nc-harmonic.pdf · For the groups listed above, we know that under some suitable restriction, we have Fourier inversion,

Documents