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Basic Functional Analysis Master 1 UPMC MM005

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Basic Functional Analysis Master 1 UPMC MM005 Basic Functional Analysis Master 1 UPMC MM005 Jean-Fran¸coisBabadjian, Didier Smets and Franck Sueur October 18, 2011 2 Contents 1 Topology 5 1.1 Basic definitions . 5 1.1.1 General topology . 5 1.1.2 Metric spaces . 6 1.2 Completeness . 7 1.2.1 Definition . 7 1.2.2 Banach fixed 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 . 11 1.4 Separability . 12 2 Spaces of continuous functions 13 2.1 Basic definitions . 13 2.2 Completeness . 13 2.3 Compactness . 14 2.4 Separability . 15 3 Measure theory and Lebesgue integration 19 3.1 Measurable spaces and measurable functions . 19 3.2 Positive measures . 20 3.3 Definition and properties of the Lebesgue integral . 21 3.3.1 Lebesgue integral of non negative measurable functions . 21 3.3.2 Lebesgue integral of real valued measurable functions . 23 3.4 Modes of convergence . 25 3.4.1 Definitions and relationships . 25 3.4.2 Equi-integrability . 27 3.5 Positive Radon measures . 29 3.6 Construction of the Lebesgue measure . 34 4 Lebesgue spaces 39 4.1 First definitions and properties . 39 4.2 Completeness . 41 4.3 Density and separability . 42 4.4 Convolution . 42 4.4.1 Definition and Young's inequality . 43 4.4.2 Mollifier . 44 4.5 A compactness result . 45 5 Continuous linear maps 47 5.1 Space of continuous linear maps . 47 5.2 Uniform boundedness principle{Banach-Steinhaus theorem . 48 5.3 Geometry of Banach spaces and identification of their dual . 49 3 6 Duality in the Lebesgue spaces and bounded measures 53 6.1 Uniform convexity and smoothness of the norm . 53 6.2 Duality in the Lebesgue spaces . 55 6.3 Bounded Radon measures . 57 7 Hilbert analysis 59 7.1 Inner product space . 59 7.2 Hilbert spaces . 61 7.3 Projection on a closed convex set . 62 7.4 Duality and weak convergence . 63 7.5 Convexity and optimization . 64 7.6 Spectral decomposition of symmetric compact operators . 65 8 Fourier series 71 8.1 Functions on the torus . 71 8.2 Fourier coefficients of L1(T; C)-functions . 71 8.3 Fourier inversion formula . 72 8.4 Functional inequalities . 75 8.5 Adaptation for T -periodic functions . 76 9 Fourier transform of integrable and square integrable functions 77 9.1 Fourier transform of integrable functions . 77 9.2 Fourier transform of L2 functions . 82 9.3 Application to the heat equation . 83 10 Tempered distributions and Sobolev spaces 87 10.1 Tempered distributions . 87 10.1.1 First definitions . 87 10.1.2 Transpose . 88 10.1.3 Fundamental solution of a differential operator with constant coefficients . 91 10.2 Sobolev spaces . 92 10.2.1 Definition . 92 10.2.2 A few properties . 92 10.2.3 Dirichlet problem . 94 4 Chapter 1 Topology In this chapter we give a few definitions of general topology including compactness and separability. One important particular case of topological spaces are the metric spaces for which most of the definitions 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 fixed point theorem, Baire's theorem and a theorem about the extension of uniformly continuous functions. 1.1 Basic definitions 1.1.1 General topology We start with recalling a few basic definitions of general topology. Definition 1.1.1 (Topology). Given a set X, we say that a subset τ of P(X) is a topology on X if 1. ; and X are in τ. 2. τ is stable by finite intersection. 3. τ is stable by union. Then we say that (X; τ) is a topological space. The elements of τ are called open sets, and their comple- mentary are the closed sets. Definition 1.1.2 (Interior and closure). Given a topological space (X; τ) and a set A ⊂ X, we define 1. the interior of A by A◦ := fx 2 A : there exists U 2 τ such that x 2 U ⊂ Ag; 2. the closure of A by A := fx 2 X : for any U 2 τ with x 2 U; then U \ A 6= ;g. We say that x 2 A is an adherent point and x 2 A◦ an interior point. Let us observe that A◦ ⊂ A ⊂ A. Definition 1.1.3 (Density). Given a topological space (X; τ) and a set A ⊂ X, we say that A is dense in X for the topology τ if A = X. Definition 1.1.4 (Limit of a sequence). Given a topological space (X; τ), we say that a sequence (xn)n2N ⊂ X converges to x in X if for any open set U 2 τ with x 2 U, there exists n0 2 N such that xn 2 U for all n > n0. Definition 1.1.5 (Continuity). Given two topological spaces (X1; τ1) and (X2; τ2), we say that a map −1 f : X1 ! X2 is continuous at x1 2 X1 if for all open set U 2 τ2 such that f(x1) 2 U, then f (U) 2 τ1. For extended real-valued functions the following notion of lower semicontinuity is weaker than conti- nuity. 5 Definition 1.1.6 (Lower semicontinuity). Given a topological space (X; τ) and x0 in X, we say that a function f : X ! R [ {−∞; +1g is lower semicontinuous at x0 if for any " > 0, there exists a neighborhood U 2 τ of x0 such that f(x) 6 f(x0) + " for all x in U. It is not difficult to check that a function is lower semicontinuous if and only if fx 2 X : f(x) > αg is an open set for every α 2 R. 1.1.2 Metric spaces An important case of topological spaces is given by metric spaces that we now introduce. Definition 1.1.7 (Distance). Given a set X, we say that a function d : X × X ! 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, z 2 X, d(x; y) 6 d(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) := fy 2 X : d(x; y) < rg (resp. B(x; r) := fy 2 X : d(x; y) 6 rg) the open (resp. closed) ball of center x and radius r. A subset is said to be bounded if it is contained in a ball of finite radius. Proposition 1.1.1. Given a metric space (X; d), the family τ of all subsets U ⊂ X such that for each x 2 U, there exists r > 0 satisfying B(x; r) ⊂ U defines 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 τ given by Proposition 1.1.1. The following statements hold: 1. A sequence (xn)n2N ⊂ X converges to x in X if and only if for every " > 0, there exists n0 2 N such that for all n > n0, then d(x; xn) < "; 2. A subset F of X is closed if and only if for every sequence (xn)n2N ⊂ F converging to x in X, then x 2 F . Proposition 1.1.3. Let (X1; d1) and (X2; d2) be two metric spaces, f : X1 ! X2 and x1 2 X1. Then the following statements are equivalent: 1. f is continuous at x1; 2. For any " > 0, there exists δ > 0 such that if x 2 X1 is such that d1(x1; x) < δ, then d2(f(x1); f(x)) < "; 3. For any sequence (xn)n2N ⊂ X1 converging to x1, the sequence (f(xn))n2N converges to f(x) in X2. Proposition 1.1.4. Given a metric space (X; d) and x0 in X. A function f : X ! R [ {−∞; +1g is lower semicontinuous at x0 if and only if for any sequence (xn)n2N ⊂ X converging to x in X, then f(x) lim inf f(xn): 6 n!1 Definition 1.1.8 (Uniform continuity). Let (X1; d1) and (X2; d2) be two metric spaces. We say that an map f : X1 ! X2 is uniformly continuous if for any " > 0, there exists δ > 0 such that if x and y 2 X1 satisfy d1(x; y) < δ, then d2(f(x); f(y)) < ". It is clear from the definitions above that uniform continuity implies continuity. We will see in Theorem 1.3.1 that the converse statement holds true when the space (X1; d1) is compact. 6 1.2 Completeness 1.2.1 Definition 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 first define the Cauchy property. Definition 1.2.1 (Cauchy sequence). Given a metric space (X; d), we say that a sequence (un)n2N ⊂ 0 X is a Cauchy sequence if for any " > 0, there exists n0 2 N such that for all n, n > n0 then d(un; un0 ) < ". Definition 1.2.2 (Completeness). A metric space (X; d) is complete if any Cauchy sequence converges in X.
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