Functional Analysis Definitions Fionn´anHoward June 2011 Contents 1 Topology 1 2 Zorn's Lemma 3 3 Nets and Separation Axioms 4 4 Banach Spaces I 5 5 Banach Spaces II 5 6 Main Theorems on Banach Spaces 6 7 Hilbert Spaces 6 1 1 Topology • Metric Space -A metric space is a set X with a function d : X × X ! [0; 1) such that for all x; y; z 2 X: 1. d(x; y) ≥ 0 with equality iff x = y 2. d(x; y) = d(y; x) 3. d(x; z) ≤ d(x; y) + d(y; z) • Open Ball - The open ball of radius r and centre x0 is given by, B(x0; r) = fx 2 X : d(x; x0) < rg: • Closed Ball - The closed ball of radius r and centre x0 is given by, B¯(x0; r) = fx 2 X : d(x; x0) ≤ rg: • Interior Point / Open Set - For a subset G ⊆ X of a metric space X, a point x 2 G is an interior point of G if 9 r > 0 such that B(x; r) ⊂ G. The set G is called an open set if every x 2 G is an interior point. • Closed Set - A set G ⊆ X of a metric space X is called a closed set if it's complement XnG is an open set. • Subspace Topology - For a subset T ⊂ X of a topological space (X; τ), we define the subspace topology τT on T to be, τT = fT \ V : V 2 τg: • Interior / Closure / Boundary - The interior Eo of a set E is the union of all open sets contained in E. The closure E¯ of a set E is the intersection of all closed sets containing E. The boundary @E of a set E is given by E¯nEo. • Continuity in a Metric Space - For metric spaces (X; dX ) and (Y; dY ), a function f : X ! Y is called continuous at x0 2 X if for each > 0, 9 δ > 0 such that, dX (x; x0) < δ =) dY (f(x); f(x0)) < . • Continuity in a Topological Space - For topological spaces (X; τX ) and (Y; τY ), a function f : X ! Y is called continuous if: −1 f (V ) 2 τX for all V 2 τY : 1 • Limit in a Metric Space - For a metric space (X; d), a sequence (xn)n=1 2 X converges to a limit l 2 X (denoted lim xn = l) if for each > 0, 9 N 2 such that, n!1 N n 2 N; n > N =) d(xn; l) < . 2 1 • Limit in a Topological Space - For a topological space (X; τ), a sequence (xn)n=1 2 X converges to a limit l 2 X if for each open set U with l 2 U, 9 N ≥ 1 such that, n 2 N; n > N =) xn 2 U: • Open Cover / Compact - For a topological space (X; τ) and a subset T ⊆ X, an open cover of T is a family U of open subsets of X such that, T ⊆ [fU : U 2 Ug: T is compact if each open cover has a finite subcover (subfamily of U which is also a cover of T ). 1 • Sequentially Compact - A subset T ⊂ X is called sequentially compact if each sequence (tn)n=1 has 1 a subsequence (tnj )j=1 that converges to some limit l 2 X. • Connected - A topological space X is called connected if the only clopen subsets of X are ; and X itself. For subsets T ⊂ X take T with the subspace topology, (T; τT ). • Base - A subfamily B ⊆ τ is called a base (for the open sets) of the topology if for each x 2 U open, there exists B 2 B such that x 2 B ⊂ U. • Second Countable - second countable if 9 a countable base. • Dense - A subset S ⊆ X is said to be dense in X if S¯ = X. • Separable - separable if 9 countable dense subset S of X. • Weaker/Stronger Topologies - Let X be a set with two topologies τ1 and τ2. τ1 is weaker than τ2 if τ1 ⊆ τ2, i.e. there are fewer open sets in the weaker topology. • Product Topology - Given two top. spaces X and Y we define the product topology to be the topology which has as its base, fU × V : U ⊂ X open;V ⊂ Y openg: ◦ • Neighbourhood - A subset N ⊆ X is a nbd of x0 if x0 2 N . • Neighbourhood System - The nbd system Ux0 at x0 is the collection of all nbds of x0. • Neighbourhood Base -A nbd base at a point x0 2 X is a subset Bx0 ⊂ Ux0 such that for each N 2 Ux0 , there exists B 2 Bx0 with B ⊆ N. • First Countable -(X; τ) is first countable at x0 2 X if 9 a countable nbd base at x0. 2 Zorn's Lemma • Partial Order - The relation ≤ is a partial order on S if for all x; y; z 2 S: 1. x ≤ x. 2. x ≤ y and y ≤ x =) x = y. 3. x ≤ y and y ≤ z =) x ≤ z . 3 • Linear Ordering -A linear ordering ≤ on a set S is a partial order with the additional property that for all x; y 2 S, either x ≤ y or y ≤ x. • Chain - For a poset (S; ≤), a chain in S is a subset C ⊆ S that is linearly ordered. • Upper Bound - For a poset (S; ≤), an elt u 2 S is an upper bound for R ⊆ S if for all x 2 R, x ≤ u. • Maximal Element - For a poset (S; ≤), m 2 S is a maximal element of S if for x 2 S with m ≤ x, x = m. • Linearly Independent - If V is a v.s. over a field K, a subset E ⊆ V is linearly independent if for distinct e1; : : : ; en 2 E and λ1; : : : ; λn 2 K, we have, λ1e1; : : : ; λnen = 0 =) λ1 = ::: = λn = 0: Span - E spans V if all v 2 V can be expressed as v = λ1e1; : : : ; λnen. Basis - E is a basis for V if it is linearly independent and spans V . 3 Nets and Separation Axioms • Hausdorff Space -(X; τ) is called a Hausdorff (or T2) space if for a; b 2 X with a 6= b, there exists U; V ⊂ X with a 2 U; b 2 V and U \ V = ;. • Directed Set (Λ; ≤) is a directed set if: 1. λ ≤ λ 2. λ1 ≤ λ2 and λ2 ≤ λ3 implies λ1 ≤ λ3 3. For all λ1; λ2 2 Λ there exists λ3 2 Λ with λ1 ≤ λ3 and λ2 ≤ λ3. • Net -A net (xλ)λ2Λ in a set X (usually a top. space) consists of: 1. a directed set (Λ; ≤) 2. a function x :Λ ! X • Converges - A net converges to x0 2 X if for each U open with x0 2 U, there exists λ0 2 Λ such that for λ ≥ λ0, xλ 2 U. • Cofinal - A subset A ⊆ Λ is called cofinal in Λ if for each λ 2 Λ there exists α 2 A with λ ≤ α. • Subnet - For a net (xλ)λ2Λ indexed by a directed set (Λ; ≤Λ), a subnet consists of two things: 1. another net (yα)α2A indexed by a set (A; ≤A) 2. a map j : A ! Λ such that (a) j is order preserving, α1 ≤A α2 =) j(α1) ≤Λ j(α2) (b) yα = xj(α) (c) j(A) is cofinal in Λ. • Cluster/Limit Point - A point x0 2 X is called a cluster/limit point of a net if for N 2 Ux0 , the set fλ 2 Λ: xλ 2 Ng is cofinal in Λ. 4 • T0 Space - if 9 U ⊆ X open with either x 2 U; y2 = U or y 2 U; x2 = U. • T1 Space - if 9 U ⊆ X open such that x 2 U; y2 = U. • Regular=T3 Space - if A ⊆ X closed, and b 2 XnA implies 9 U; V ⊆ X open such that A ⊆ U; b 2 V and U \ V = ;. T3 if it is both regular and Hausdorff. • Normal=T4 Space - if A; B ⊆ X closed, A \ B = ; implies 9 U; V ⊆ X open such that A ⊆ U; B ⊆ V an U \ V = ;. T4 if it is both normal and Hausdorff. • Completely Regular=T 1 Space - A topological space is called completely regular if for A ⊂ X 3 2 closed with b 2 XnA, there exists continuous f : X ! R with f(a) = 0 for all a 2 A and f(b) = 1. T 1 if it is both completely regular and Hausdorff. 3 2 4 Banach Spaces I • Norm -A norm on a v.s. E over K is a function kxk : E ! [0; 1) such that for all x; y 2 E: 1. kx + yk ≤ kxk + kyk 2. kλxk = jλjkxk for all λ 2 K 3. kxk = 0 =) x = 0 • Seminorm -A seminorm (denoted p(x) instead of kxk) is a norm that doesn't satisfy 3 above. Non-zero elts can have length 0. 1 • Cauchy Sequence - Note that this definition requires a metric. A sequence (xn)n=1 in a metric space is called a Cauchy sequence if for each > 0, 9N such that, n; m ≥ N =) d(xn; xm) < . • Complete - A metric space is called complete if every Cauchy sequence converges to a limit. • Uniformly Continuous - A function f : X ! Y is called uniformly continuous if for each > 0, 9 δ > 0 such that, d(x1; x2) < δ =) d(f(x1); f(x2)) < . • Banach Space - A normed space (E; k · k) over K is called a Banach space over K if E is complete in the metric arising from the norm.