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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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  • General Topology

    General Topology

    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).