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Functional Analysis 1 MATHEMATISCHES INSTITUT PROF. DR. PETER MÜLLER Summer Term 2013 Lecture Course Functional Analysis Typesetting by Kilian Lieret and Marcel Schaub If you find mistakes, I would appreciate getting a short mail from you to marcel.schaub [at] campus.lmu.de. Thanks! Version of April 10, 2014 Contents 1 Topological and metric spaces5 1.1 Topological spaces: basics............................5 1.2 Limits and continuity..............................7 1.3 Metric spaces...................................8 1.4 Example: sequence spaces `p .......................... 14 1.5 Compactness................................... 16 1.6 Example: spaces of continuous functions.................... 19 1.7 Baire's Theorem................................. 23 2 Banach and Hilbert spaces 26 2.1 Vector spaces................................... 26 2.2 Banach spaces................................... 27 2.3 Linear operators................................. 30 2.4 Linear functionals and dual space........................ 35 2.5 Hilbert spaces................................... 37 3 Measures, integration and Lp-spaces 47 3.1 Measures..................................... 47 3.2 Integration.................................... 49 3.3 Lp-spaces..................................... 53 3.4 Decomposition of Measures........................... 66 4 The cornerstones of functional analysis 69 4.1 Hahn-Banach theorem.............................. 69 4.2 Three consequences of Baire's theorem..................... 72 4.3 (Bi)-Dual spaces and weak topologies...................... 76 5 Bounded operators 85 5.1 Topologies on the space of bounded linear operators............. 85 5.2 Adjoint operators................................. 88 5.3 The spectrum................................... 91 5.4 Compact operators................................ 95 5.5 Fredholm alternative for compact operators.................. 98 3 Introductory remarks Functional analysis is... a child of linear algebra and analysis • a theory of infinite-dimensional vector spaces • Functional analysis has lots of applications partial differential equations (PDE's) • approximation theory • numerical maths • probability theory • quantum mechanics (functional analysis is its language!) • 4 1 Topological and metric spaces 1.1 Topological spaces: basics 1.1 Definition. Let X be a set. (X) is a topology iff T ⊆ P (1) ?;X . 2 T (2) is closed under arbitrary unions (i.e. if I is an arbitrary index set and for every T α I let a set A be given. Then 2 α 2 T [ A α 2 T α I 2 holds.). (3) is closed under finite intersections (i.e. if n N and A1;:::;An , then T 2 2 T n \ A k 2 T k=1 holds.). (X; ) is called topological space (often just X) • T A (X) is open iff A . • 2 P 2 T Let ; be topologies on X. is finer than iff and coarser than iff • T1 T2 T1 T2 T1 ⊇ T2 T2 . T1 ⊆ T2 1.2 Examples. (a) Indiscrete topology: = ?;X . T f g (b) Discrete topology: = (X). T P (c) Euclidean (or standard) topology on Rn, n N: A Rn is open iff x A " > 0 2n ⊆ 8 2 9 such that B"(x) A, where B"(x) := y R : x y < " is the Euclidean ball of ⊆ f 2 j − j g radius " > 0 about x Rn. 2 Induced topology on subsets 1.3 Definition. Let (X; ) be a topological space, A (X) (not necessarily open!). T 2 P (A) is the relative topology on A iff TA ⊆ P := B A : C with B = C A : TA f ⊆ 9 2 T \ g 1.4 Remark. (a) is topology on A. TA (b) If A = and B then it may happen that B = . 2 T 2 TA 2 T Example. Let X = R with standard topology, A = [0; 1]. Then B := [0; 1=2[ A 2 T but B = . 2 T 1.5 Definition. Let X be a topological space, A X, x X. ⊆ 2 (a) A is closed iff Ac := X A . n 2 T 5 (b) U X (not necessarily open) is a neighbourhood of x iff A such that x A and ⊆ 9 2 T 2 A U. ⊆ (c) X is a Hausdorff space iff for all x; y X, x = y, there exist neighbourhoods U of x 2 6 x and Uy of y such that Ux Uy = ?. \ (d) x is a limit point of A (or accumulation point) iff for all neighbourhoods U of x U A = ?: \ 6 Note: Every point of A is also a limit point according to this definition. (e) x is an interior point of A iff there exists a neighbourhood U of x such that U A. ⊆ (f) x is a boundary point of A iff for every neighbourhood U of x: U A = ? and \ 6 U Ac = ?. \ 6 Boundary of A: @A := x X : x boundary point of A . f 2 g (g) Interior of A: A˚ := A @A closure of A: A := A @A = x X : n [ f 2 x limit point of A . g (h) A is dense in X iff X = A. 1.6 Lemma. Let X be a topological space, A X. ⊆ (a) A is open x A : x is an interior point of A. () 8 2 (b) A is closed A = A. () (c) A; @A are closed. Proof. See Problem 1. 1.7 Definition. Let (X; ) be a topological space, a family of open sets. T B ⊆ T (a) is a base for iff consists of unions of sets from . B T T B (b) is a subbase for iff finite intersections of sets from form a base. B T B (c) is a neighbourhood base at x iff every N is a neighbourhood of x and for N ⊆ T 2 N every neighbourhood U of x there exists N with N U. 2 N ⊆ 1.8 Remark. (a) Let (X). Then there exists a topology on X such that is a S ⊆ P T S subbase for and is the coarsest topology containing . Jargon: is generated T T S T by . S (b) Example: Rn with standard topology. Let x Rn. 2 (i) B1=k(x): k N is a neighbourhood base at x. f 2 g n (ii) B1=k(q): k N; q Q is a base for the standard topology (see the proof of f 2 2 g Thm. 1.19 later). 6 1.9 Definition. Let J = ? be an arbitrary index set. For every j J let (Xj; j) be a 6 2 T topological space. is the product topology on the Cartesian product space T [ X := f : J X with f(j) X j ! j 2 j j J j J ×2 2 iff has the base T Aj : Aj j j J; Aj = Xj for at most finitely many j's : j J 2 T 8 2 6 ×2 1.10 Remark. If J is finite, then the condition \A = X for at most finitely many j's" j 6 j can be dropped. 1.11 Definition. Let X be a topological space. (a) X is separable iff A X countable with A = X. 9 ⊆ (b) X is 1st countable iff every x X has a countable neighbourhood base. 2 (c) X is 2nd countable iff there exists a countable (sub-)base for the topology. [Note: countable base countable sub base (see Problem T2).] () 1.12 Theorem. Let X be a topological space. Then X is 2ndcountable = X is 1stcountable and separable. ) Proof. Let be a countable base for the topology. B Let x X and := B : x B . Then is a neighbourhood base and • 2 Nx f 2 B 2 g Nx countable (hence 1st countable). ? = B choose xB B and let A := xB : ? = B . We claim A is •8 6 2 B 2 f 6 2 Bg countable (trivial) and A = X. For all x X and neighbourhoods U of x there 2 exists C such that x C U. C is a union from sets in , so there exists 2 T 2 ⊆ B B such that x B U. On the other hand x B means x U and x A 2 B 2 ⊆ B 2 B 2 B 2 implies A U = ?. \ 6 1.2 Limits and continuity 1.13 Definition. Let X be a topological space and (xn)n N X be a sequence. 2 ⊆ (xn)n converges to x X iff for every neighbourhood U of x there exists n0 N and for 2 2 all n n0: xn U. ≥ 2 n Notations: limn xn = x or xn !1 x. !1 −−−! 1.14 Remark. (a) convergence is harder for finer topologies. (b) X Hausdorff = limits are unique (see Problem T3). ) 1.15 Definition. Let (X; ) and (Y; ) be topological spaces and let f : X Y . TX TY −! n n (a) f is sequentially continuous iff x !1 x (in X) implies f(x ) !1 f(x) (in Y ). n −−−! n −−−! (b) f is continuous iff for every A : f 1(A) (f 1(A) is the inverse image). 2 TY − 2 TX − (c) f is open iff A : f(A) = y Y : x A : y = f(x) . 8 2 TX f 2 9 2 g 2 TY 7 (d) f is a homeomorphism iff f is bijective, open and continuous (bijection compatible with topological structure). 1.16 Theorem. Let X; Y be topological spaces and f : X Y a map. Then −! (a) f is continuous = f is sequentically continuous. ) (b) f is sequentially continuous and X is first countable = f is continuous. ) n Proof. (a) Let x !1 x in X. Let V Y be a neighbourhood of f(x). W.l.o.g.1 assume n −−−! ⊆ V open (see below). Set U := f 1(V ). U is open (because f is continuous) and x U − 2 = U is a neighbourhood of x and we can apply the definition of convergence to U ) = n0 N : n n0 : xn U. But that also means n0 N: n n0 : f(xn) V . ) 9 2 8 ≥ 2 9 2 8 ≥ 2 If V is not open there exists an open subset V V with x V and one can repeat 0 ⊆ 2 0 the same argument with V0 instead of V .
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