Basic Functional Analysis

Master 1 UPMC

MM005

Jean-Fran cois Babadjian, Didier Smets and Franck Sueur

June 30, 20112Contents

1 Topology 5

1.1 Basic de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Banach xed point theorem for contraction mapping . . . . . . . . . . . . . . . . 7

1.2.3 Baire’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Extension of uniformly continuous functions . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Banach spaces and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Spaces of continuous functions 11

2.1 Basic de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Measure theory and Lebesgue integration 17

3.1 Measurable spaces and measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Positive measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 De nition and properties of the Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Lebesgue integral of non negative measurable functions . . . . . . . . . . . . . . . 19

3.3.2 Lebesgue in of real valued measurable functions . . . . . . . . . . . . . . . . 21

3.4 Modes of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 De nitions and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Equi-integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Positive Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Construction of the Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Lebesgue spaces 37

4.1 First denitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Density and separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4.1 De nition and Young’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.2 Molli er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 A compactness result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Continuous linear maps 45

5.1 Space of continuous linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Uniform boundedness principle{Banach-Steinhaus theorem . . . . . . . . . . . . . . . . . 46

5.3 Geometry of Banach spaces and identi cation of their dual . . . . . . . . . . . . . . . . . 47

36 Duality in the Lebesgue spaces and bounded measures 51

6.1 Uniform convexity and smoothness of the norm . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Duality in the Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 Bounded Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Hilbert analysis 57

7.1 Inner product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3 Projection on a closed convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4 Duality and weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.5 Convexity and optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.6 Spectral decomposition of symmetric compact operators . . . . . . . . . . . . . . . . . . . 63

8 Fourier series 69

8.1 Functions on the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

18.2 Fourier coe cients of L (T;C)-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.3 Fourier inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.4 Functional inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.5 Adaptation for T -periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9 Fourier transform of integrable and square integrable functions 75

9.1 Fourier of in functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

29.2 F transform of L functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9.3 Application to the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

10 Tempered distributions and Sobolev spaces 85

10.1 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

10.1.1 First de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

10.1.2 Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10.1.3 Fundamental solution of a di erential operator with constant coe cients . . . . . . 89

10.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10.2.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10.2.2 A few properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10.2.3 Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

46

Chapter 1

Topology

In this chapter we give a few de nitions of general topology including compactness and separability. One

important particular case of topological spaces are the metric spaces for which most of the de nitions

can be rephrased in term of sequences. We will also introduce the notion of completeness and we give

three theorems using it in a crucial way: the Banach xed point theorem, Baire’s theorem and a theorem

about the extension of uniformly continuous functions.

1.1 Basic de nitions

1.1.1 General topology

We start with recalling a few basic de nitions of general topology.

De nition 1.1.1 (Topology). Given a set X, we say that a subset ofP(X) is a topology on X if

1. ; and X are in .

2. is stable by nite intersection.

3. is stable by union.

Then we say that (X;) is a topological space. The elements of are called open sets, and their comple-

mentary are the closed sets.

De nition 1.1.2 (Interior and closure). Given a topological space (X;) and a setAX, we de ne

1. the interior of A by A :=fx2A : there exists U2 such that x2UAg;

2. the closure of A by A :=fx2A : for any U2 with x2U; then U\A =;g.

We say that x2A is an adherent point and x2A an interior point.

Let us observe that A AA.

De nition 1.1.3 (Density). Given a topological space (X;) and a set AX, we say that A is dense

in X for the topology if A =X.

De nition 1.1.4 (Limit of a sequence). Given a topological space (X;), we say that a sequence

(x ) X converges to x in X if for any open set U2 with x2 U, there exists n 2N such thatn n2N 0

x 2U for all n>n .n 0

De nition 1.1.5 (Continuity). Given two topological spaces (X ; ) and (X ; ), we say that a map1 1 2 2

1f :X !X is continuous at x 2X if for all open set U2 such that f(x )2U, then f (U)2 .1 2 1 1 2 1 1

For extended real-valued functions the following notion of lower semicontinuity is weaker than conti-

nuity.

5De nition 1.1.6 (Lower semicontinuity). Given a topological space (X;) and x in X, we say0

that a function f : X!R[f 1 ; +1g is lower semicontinuous at x if for any " > 0, there exists a0

neighborhood U2 of x such that f(x)6f(x ) +" for all x in U.0 0

It is not di cult to check that a function is lower semicontinuous if and only if fx2X : f(x)>g

is an open set for every 2R.

1.1.2 Metric spaces

An important case of topological spaces is given by metric spaces that we now introduce.

+De nition 1.1.7 (Distance). Given a set X, we say that a function d :XX!R is a distance on

X if

1. d(x;y) = 0 if and only if x =y;

2. d(x;y) =d(y;x);

3. For any x, y, z2X, d(x;y)6d(x;z) +d(y;z).

Then we say that (X;d) is a metric space.

For any x in X and any r> 0, we denote

B(x;r) :=fy2X : d(x;y)<rg (resp. B(x;r) :=fy2X : d(x;y)6rg)

the open (resp. closed) ball of center x and radiusr. A subset is said to be bounded if it is contained in

a ball of nite radius.

Proposition 1.1.1. Given a metric space (X;d), the family of all subsets UX such that for each

x2U, there exists r> 0 satisfying B(x;r)U de nes a topology on X.

The following propositions, whose proofs are left to the reader, highlight the role of the sequences in

metric spaces.

Proposition 1.1.2. Given a metric space (X;d) and the topology given by Proposition 1.1.1. The

following statements hold:

1. A sequence (x ) X converges to x in X if and only if for every " > 0, there exists n 2Nn n2N 0

such that for all n>n , then d(x;x )<";0 n

2. A subset F of X is closed if and only if for every sequence (x ) F converging to x in X,n n2N

then x2F .

Proposition 1.1.3. Let (X ;d ) and (X ;d ) be two metric spaces, f :X !X and x 2X . Then1 1 2 2 1 2 1 1

the following statements are equivalent:

1. f is continuous at x ;1

2. For any"> 0, there exists> 0 such that ifx2X is such thatd (x ;x)<, thend (f(x );f(x))<1 1 1 2 1

";

3. For any sequence (x ) X converging to x , the sequence (f(x )) converges to f(x) inn n2N 1 1 n n2N

X .2

Proposition 1.1.4. Given a metric space (X;d) and x in X. A function f :X!R[f 1 ; +1g is0

lower semicontinuous at x if and only if for any sequence (x ) X converging to x in X, then0 n n2N

f(x)6 lim inff(x ):n

n!1

De nition 1.1.8 (Uniform continuity). Let (X ;d ) and (X ;d ) be two metric spaces. We say that1 1 2 2

an map f : X ! X is uniformly continuous if for any " > 0, there exists > 0 such that if x and1 2

y2X satisfy d (x;y)<, then d (f(x);f(y))<".1 1 2

It is clear from the de nitions above that uniform continuity implies continuity. We will see in

Theorem 1.3.1 that the converse statement holds true when the space (X ;d ) is compact.1 1

66

1.2 Completeness

1.2.1 De nition

Completeness is an important notion in general topology and in functional analysis because it enables

one to characterize converging sequences without the knowledge of their limit. We rst de ne the Cauchy

property.

De nition 1.2.1 (Cauchy sequence). Given a metric space (X;d), we say that a sequence (u ) n n2N

0X is a Cauchy sequence if for any " > 0, there exists n 2 N such that for all n, n > n then0 0

0d(u ;u )<".n n

De nition 1.2.2 (Completeness). A metric space (X;d) is complete if any Cauchy sequence converges

in X.

Let us give as a rst example the set R endowed with the usual metric d(x;y) :=jx yj. It is also

useful to notice that a closed subset of a complete metric space is complete.

1.2.2 Banach xed point theorem for contraction mapping

An important application of the notion of completeness is given by the following theorem.

Theorem 1.2.1 (Banach, Picard). Given a complete metric space (X;d) and f : X! X. Assume

that f is a contraction, i.e. that there exists a constant 2 (0; 1) such that for all x and y2 X, then

d(f(x);f(y))6d(x;y). Then there exists a unique xed point x 2X such that f(x ) =x .

Proof. Let x 2X and let (x ) the associated sequence de ned by the relation0 n n2N

x =f(x ): (1.1)n+1 n

By iteration we have

nd(x ;x )6 d(x ;x ):n+1 n 1 0

0For any n >n,

0 0n n n n nX X n+k 1d(x 0;x )6 d(x ;x )6d(x ;x ) 6 d(x ;x ):n n n+k n+k 1 1 0 1 0

1

k=1 k=1

Therefore (x ) is a Cauchy sequence, and by completeness, it converges to an element x 2X. Sincen n2N

f is continuous, we have f(x ) = x by passing to the limit in (1.1). The uniqueness follows from the

contraction assumption.

The previous theorem is useful in the proof of the Cauchy-Lipschitz theorem in the theory of ordinary

di erential equations, and also in proof the local invertion theorem.

1.2.3 Baire’s theorem

The following theorem was proved by Baire in his 1899 doctoral thesis.

Theorem 1.2.2 (Baire). In a complete metric space, every intersection of countable collection of dense

open sets is dense.

Proof. Let (X;d) be a complete metric space andfU g be a sequence of dense open sets with then n2N

property that U = X for each n2N. To prove the result it su ces to show that for any open balln

B in X, then B\ (\ U ) =;. Since U is dense in X, there exists x in X and r > 0 such thatn2N n 0 0 0

B(x ;r )B\U . By iteration, using the fact that every sets U are dense inX, we obtain that there0 0 0 n

exists a sequence (x ) X a sequence (r ) of positive real numbers with r <r =2 such thatn n2N n n2N n n 1

nB(x ;r )B(x ;r )\U . Since fork>n,x 2B(x ;r ) withr <r =2 , the sequence (x )n n n 1 n 1 n k n n n 0 n n2N

has the Cauchy property. By completeness, there exists x2 X such that (x ) converges to x. Wen n2N

conclude by observing that x2B\ (\ U ) since x2B(x ;r )U \B for any n2N.n2N n n n n

71.2.4 Extension of uniformly continuous functions

Theorem 1.2.3 (Extension of uniformly continuous functions). Given two metric spaces (X ;d )1 1

and (X ;d ), the latter being complete, a dense subset Y of X , and an map f : Y ! X which is2 2 1 2

uniformly continuous. Then there exists a unique uniformly continuous extension g :X !X of f.1 2

Proof. The uniqueness is straightforward: indeed for any x2X , since Y is dense in X , there exists a1 1

sequence (x ) Y which converges to x. If g :X !X is a uniformly continuous extension of f,n n2N 1 2

then g(x) must be the limit of the sequence (f(x )) .n n2N

Now to prove the existence of such an extension, observe that the sequence (f(x )) has the Cauchyn n2N

property, since (x ) has the Cauchy property (because it converges) and f is uniformly continuous.n n2N

Since (X ;d ) is complete, it yields that the sequence (f(x )) converges to an element z in X .2 2 n n2N 2

0This z does not depend on the choice of the (x ) in X . Indeed, if (x ) is anothern n2N 1 n2Nn

0sequence converging to x, then d (x ;x )! 0 when n! +1, so that, since f is uniformly continuous,1 n n

0 0d (f(x );f(x ))! 0 whenn! +1. Hence the sequence (f(x )) converges toz as well. Therefore,2 n n2Nn n

it makes sense to de ne g(x) :=z.

Let us now prove that g is uniformly continuous. Let " > 0. Since f is uniformly continuous there

0 0 0exists > 0 such that for any x, x 2 Y with d (x;x ) < , there holds d (f(x);f(x )) < "=3. Let1 2

0 0 0 0 0y and y 2 X with d (y;y ) < =3. There exists x, x 2 Y such that d (x;y) < =3, d (x;y ) <1 1 1 1

0 0=3, d (f(x);g(y)) < "=3 and d (f(x );g(y )) < "=3. Therefore, thanks to the triangle inequality, we2 2

0 0have d (x;x ) < and therefore d (f(x);f(x )) < "=3. Using again the triangle inequality, we get1 2

0d (g(y);g(y ))<". Hence g is uniformly continuous.2

Some typical applications of the extension of a uniformly continuous function can be found in the study

of the convolution product (see Corollary 4.4.1), and in the proof of the inverse Fourier transformation

formula (section 9).

1.2.5 Banach spaces and algebra

Let us recall a few de nitions:

De nition 1.2.3 (Normed vector space). A normed vector space overR is a pair (V;kk) where V is

a vector space andkk is a norm on X that is a function from X toR satisfying+

1. kuk = 0 if and only u = 0 (positive de niteness),

2. for any u in V , for any 2R,k u k =jjkuk (positive homogeneity),

3. for any u, v in V ,ku +vk6kuk +kvk (triangle inequality or subadditivity).

We can easily associate a distance to the norm of a normed vector space, through the formula

d(u;v) :=ku vk. If the topology de ned by this distance is complete then we say that ( V;kk) is a

Banach space. If in additionV is an associative algebra whose multiplication law is compatible with the

norm in the sense thatkuvk6kukkvk for anyu,v inV , then we say that (V;kk) is a Banach algebra.

In a normed vector space, completeness can be characterized thanks to the series.

Proposition 1.2.1. Let (V;kk) be a normed vector space. Then (V;kk) is a Banach space if and only

if the series normally converging actually converge in (V;kk).

P

Let us recall that a for sequence (u ) in V , we say that the series u is normally converging inn n nnP

V if ku k converges inR.nn

1.3 Compactness

Several notions of compactness are available. The following one can be formulated in a general setting.

De nition 1.3.1 (Compactness). We say that a topological space (X;) is compact if any open cover

has a nite subcover, i.e. for every arbitrary collection fUg of open subsets of X such that X i i2I

[ U , there is a nite subset JI such that X[ U .i2I i i2J i

8It is a good exercise to prove the following theorem in order to understand the power of the previous

de nition.

Theorem 1.3.1 (Heine). Every continuous image of a compact set is compact. Moreover a continuous

function on a compact set is uniformly compact.

In a metric space, compactness can be formulated in terms of sequences. Let us rst recall a few

facts about the notion of limit points. Let S be a subset of a topological space X. We say that a point

x2X is a limit point of S if every open set containing x also contains a point of S other than x itself.

In a metric space, it is equivalent to requiring that every neighbourhood of x contains in nitely many

points of S.

Let us also de ne what we mean by a totally bounded space.

De nition 1.3.2 (Totally boundedness). We say that a metric space (X;d) is totally bounded if for

every "> 0, there exists a nite cover of X by open balls of radius less than ".

Since for every "> 0,[ B(x;") is an open cover of X, it follows from De nitions 1.3.1 and 1.3.2x2X

that a compact metric space is totally bounded.

Proposition 1.3.1. A metric space is compact if and only if every sequence has a limit point.

Proof. We start by proving the necessary condition. Let us assume by contradiction that (x ) isn n2N

a sequence in a compact metric space X without any limit point. Then for every y in X, there exists

r(y)> 0 such that the ballB(y;r(y)) contains only nitely many elements of the sequence. The collection

of these balls is a open cover of X, from which we extract a nite subcover. This would yield that the

sequence is in the union of a nite number of balls each of them containing only nitely many elements

of the sequence which is absurd.

Let us now prove the su cient condition. We therefore consider a metric space X such that every

sequence has a limit point. We rst prove that X is totally bounded. Proceeding by contradiction,

it would yield the existence of some " > 0 and some sequence (x ) such that for any m, n2 N,n n2N

d(x ;x )>". Such a sequence cannot have a limit point, which is a contradiction. Hence X is totallym n

bounded.

Let us now prove thatX is compact. We consider an open coverfUg ofX. We de ne the mappingi i2I

R :x2X7!R(x) := supfr> 0 : 9i2I with B(x;r)Ug> 0;i

which is lower semicontinuous. Indeed, if not, there would exist a sequence (x ) X converging ton n2N

x in X such that

R(x)> lim infR(x );n

n!1

0 0and we could choose and such that R(x) > > > lim inf R(x ). Let i2 I be such thatn n

0 0B(x; )U . Since the sequence (x ) converges to x, for n large enough B(x ;)B(x; )U ,i n n2N n i

which is against the fact that > lim inf R(x ).n n

Let us now consider " := inf R(x) and (x ) a minimizing sequence for R, that is such thatx2X n n2N

lim R(x ) =":n

n!1

Then by assumption, the sequence (x ) has a limit point that we call x. Since R is lower semi-n n2N

continuous, we have

0<R(x)6 lim infR(x ) =":n

n!1

We already know that X is totally bounded. Thus for that " there exist x ;:::;x such that1 n

n[

X B(x ;"):i

i=1

By de nition of ", for any i = 1;:::;n one has " 6 R(x ), and thus there exists j 2 I such thati i

B(x ;")2U and nallyi ji

n[

X Uji

i=1

which gives a nite subcover of X.

9