Cours-MM005

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• Publié par: Orgoi
• Nombre de lectures: 45
• Langue: English

Extrait

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 su