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Functional Analysis Functional Analysis Peter Philip∗ Lecture Notes Originally Created for the Class of Summer Semester 2017 at LMU Munich Includes Subsequent Corrections and Revisions† September 19, 2021 Contents 1 Topological Vector Spaces 3 1.1 BasicDefinitionsandProperties . ... 3 1.2 Finite-DimensionalSpaces . .. 14 1.3 Metrization .................................. 16 1.4 Boundedness, Cauchy Sequences, Continuity . ...... 21 1.5 SeminormsandLocalConvexity. 25 1.6 FurtherExamples............................... 33 2 Main Theorems 37 2.1 BaireCategory ................................ 37 2.2 Uniform Boundedness Principle, Banach-Steinhaus Theorem ....... 46 2.3 OpenMappingTheorem ........................... 52 2.4 ClosedGraphTheorem............................ 56 2.4.1 TheTheorem ............................. 56 2.4.2 ApplicationtoSequenceSpaces . 59 ∗E-Mail: [email protected] †Resources used in the preparation of this text include [Els07, Rud73, Wer11]. 1 CONTENTS 2 3 Convexity 63 3.1 Hahn-BanachTheorems ........................... 63 3.2 WeakTopology,WeakConvergence . 69 3.3 Banach-Alaoglu ................................ 78 3.4 ExtremePoints,Krein-Milman. .. 85 4 Duality, Representation Theorems 89 4.1 GeneralNormedSpace,AdjointOperators . .. 89 4.2 Hilbert Space, Riesz Representation Theorem I . ..... 97 4.3 ComplexMeasures,Radon-NikodymTheorem . 116 4.4 Lp-Spaces,RieszRepresentationTheoremII . 126 4.5 Borel Measures on Locally Compact Hausdorff Spaces and Riesz Repre- sentationTheoremsIIIandIV. 132 A Exhaustion by Compact Sets 155 B InterchangingDerivativeswithPointwiseLimits 156 CWeierstrassApproximationTheorem 156 D Topological Invariants 159 E Orthogonalization 160 F Tietze-Urysohn Theorem 161 G Partition of Unity 167 References 168 1 TOPOLOGICAL VECTOR SPACES 3 1 Topological Vector Spaces 1.1 Basic Definitions and Properties As in Analysis I–III, we write K to denote R or C. The central topic of (linear) Functional Analysis is the investigation and representation of continuous linear functionals, i.e. of continuous linear functions f : X K, where X is a vector space over K. To know what continuity of f means, we need−→ to specify topologies on X and K. On K, we will always consider the standard topology (induced by ), unless another topology is explicitly specified. While one will, in general, want to study|·| many different vector spaces X with many different topologies , should at least be compatible with the linear structure on X, giving rise to the followingT T definition: Definition 1.1. Let X be a vector space over K and let be a topology on X. Then the topological space (X, ) is called a topological vectorT space if, and only if, addition and scalar multiplication areT continuous, i.e. if, and only if, the maps +: X X X, (x,y) x + y, (1.1a) × −→ 7→ : K X X, (λ,x) λx, (1.1b) · × −→ 7→ are continuous (with respect to the respective product topology). Example 1.2. (a) Every normed vector space (X, ) over K is a topological vector k · k space: Let (xk)k N and (yk)k N be sequences in X with limk xk = x X and ∈ ∈ →∞ ∈ limk yk = y X. Then limk (xk + yk)= x + y by [Phi16b, (2.20a)], showing →∞ ∈ →∞ continuity of addition. Now let (λk)k N be a sequence in K such that limk λk = ∈ + →∞ λ K. Then ( λk )k N is bounded by some M R0 and ∈ | | ∈ ∈ λ x λx λ x λ x + λ x λx M x x + λ λ x 0 for k , k k k− k ≤ k k k− k k k k − k≤ k k− k | k− | k k → → ∞ showing limk (λkxk) = λx and the continuity of scalar multiplication. We will →∞ see in Sec. 1.2 below that, for dim X < , the norm topology on X is the only T ∞ 1 topology on X that makes X into a topological vector space (but cf. (b) below). (b) Let X be a vector space over K. With the indiscrete topology, X is always a topological vector space (the continuity of addition and scalar multiplication is trivial). If X = 0 , then the indiscrete space is not T1 and, hence, not metrizable (cf. [Phi16b, Sec.6 { 3.1]).} If X = 0 , then, with the discrete topology, X is never a topological vector space: While6 addition{ } is continuous (since X X is also discrete), scalar multiplication is not: Let 0 = x X. Then, while x× X is open, the 1 6 ∈ { } ⊆ preimage P := ( )− ( x ) is not open in K X: Let(λ,y) P , i.e. λy = x. If P were open, then there· had{ to} be an open neighborhood× O of λ∈ such that O y P , × { }⊆ in contradiction to λ being the unique element of K such that λy = x. 1 TOPOLOGICAL VECTOR SPACES 4 (c) Let (Ω, ,µ) be a measure space. Then, for each 0 < p , both p(µ) and Lp(µ) areA topological vector spaces: First, let 1 p . Then≤ ∞ Lp(µ) isL a special case of (a), and, for, p(µ) one observes that the≤ arguments≤ ∞ of (a) still work if the norm is replaced by aL seminorm (since seminormed spaces are still first countable, cf. [Phi16b, Th. 2.8]). If there exists a nonempty µ-null set, then p(µ) is not L T1 (in particular, not metrizable), cf. [Phi17, Def. and Rem. 2.41]. Now consider 0 <p< 1. We know from [Phi17, Def. and Rem. 2.41] that p(µ) is a pseudometric L space, where the pseudometric dp is defined by 1/p d : p(µ) p(µ) R+, d (f,g) := N p(f g), N (f) := f p dµ , p L ×L −→ 0 p p − p | | ZΩ and Lp(µ) is the corresponding (factor) metric space. Like metric spaces, pseu- dometric spaces are first countable and we can show continuity using sequences p according to [Phi16b, Th. 2.8]. Let (fk)k N and (gk)k N be sequences in (µ) with p ∈ p ∈ L limk fk = f (µ) and limk gk = g (µ). Then →∞ ∈L →∞ ∈L [Phi17, (2.51a)] d (f + g ,f + g)= N p f + g (f + g) N p(f f)+ N p(g g) p k k p k k − ≤ p k − p k − = dp(fk,f)+ dp(gk,g) 0 for k , → → ∞ showing continuity of addition. Now let (λk)k N be a sequence in K such that ∈ + limk λk = λ K. Then ( λk )k N is bounded by some M R0 and →∞ ∈ | | ∈ ∈ d (λ f ,λf) d (λ f ,λ f)+ d (λ f,λf) p k k ≤ p k k k p k p p = λ f λ f dµ + λ f λf dµ k k − k k − ZΩ ZΩ M p d (f ,f)+ λ λ p N p(f) 0 for k , ≤ p k | k − | p → → ∞ showing limk (λkfk) = λf and the continuity of scalar multiplication. As for →∞ p p 1, if there exists a nonempty µ-null set, then (µ) is not T1 (in particular, not≥ metrizable), again cf. [Phi17, Def. and Rem. 2.41].L We will see in Ex. 1.11(b) that, for 0 <p< 1, balls in Lp([0, 1], 1,λ1), where 1 denotes the usual σ-algebra of Lebesgue-measurable sets and λ1 denotesL LebesgueL measure, are not convex. In particular, the metric d is not generated by any norm on Lp([0, 1], 1,λ1). p L (d) Consider X := KR = (R, K), i.e. the set of functions f : R K, with the product topology (i.e.F the topology of pointwise convergence).−→ We know from T [Phi16b, Ex. 1.53(c)] that is not metrizable (however, is T2 by [Phi16b, Prop. 3.5(b)]). We show that (X,T ) is a topological vector spaceT over K: According T to [Phi16b, Ex. 2.12(c)], we have to show that, for each f,g : R K, each −→ 1 TOPOLOGICAL VECTOR SPACES 5 λ K, and each t R, (f,g) f(t)+ g(t) is continuous at (f,g), (λ,f) λf(t) is∈ continuous at (λ,f∈ ). Due7→ to the continuity of the maps + : K K7→ K, + × −→ : K K K, given ǫ R , there exist neighborhoods Uf , Ug, Uλ of f(t), g(t), and· λ×, respectively,−→ such that∈ z + w Bǫ f(t)+ g(t) , µz Bǫ λf(t) . (z,w) ∀Uf Ug ∈ (µ,z) ∀Uλ Uf ∈ ∈ × ∈ × 1 1 Letting Vf := πt− (Uf ), Vg := πt− (Ug), we obtain f˜(t)+˜g(t) Bǫ f(t)+ g(t) , (f,˜ g˜) ∀V Vg ∈ ∈ f × proving continuity of addition on X. We also have µf˜(t) Bǫ λf(t) , (µ,f˜) ∀U V ∈ ∈ λ× f proving continuity of scalar multiplication. — In certain situations, the following notation has already been used in both Analysis and Linear Algebra: Notation 1.3. Let X be a vector space over K, let (X) (where (X) denotes the power set of X), A, B X, x X, and λ K. DefineA⊆P P ⊆ ∈ ∈ x + A := x + a : a A , (1.2a) { ∈ } A + B := a + b : a A, b B , (1.2b) { ∈ ∈ } λA := λa : a A , (1.2c) { ∈ } x + := x + A : A . (1.2d) A { ∈ A} Note that, in general, the familiar arithmetic laws do not hold for set arithmetic: For example, if X = 0 , then X X = X = 0 ; if 0 = x X, A := x,x , then 0 A + A, but6 0 /{ 2}A, i.e. 2A =− A + A. 6 { } 6 ∈ {− } ∈ ∈ 6 Proposition 1.4. Let (X, ) be a topological vector space over K. T (a) For each a X and each λ K 0 , the maps ∈ ∈ \ { } T ,M : X X, T (x) := x + a, M (x) := λx, (1.3) a λ −→ a λ 1 1 − are homeomorphisms, where Ta− = T a, Mλ− = Mλ 1 . − 1 TOPOLOGICAL VECTOR SPACES 6 (b) is both translation-invariant and scaling-invariant, i.e. the following holds for eachT O X: ⊆ O open O + a open λO open . ⇔ a ∀X ⇔ λ K∀ 0 ∈ ∈ \{ } (c) Let x,a X, U X, λ K 0 .
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