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Locally Convex Vector Spaces I: Basic Local Theory

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Locally Convex Vector Spaces I: Basic Local Theory LCVS I c Gabriel Nagy Locally Convex Vector Spaces I: Basic Local Theory Notes from the Functional Analysis Course (Fall 07 - Spring 08) Convention. Throughout this note K will be one of the fields R or C, and all vector spaces are over K. Definition. A locally convex vector space is a pair (X , T) consisting of a vector space X and linear topology T on X , which is locally convex, in the sense that: (lc) every x ∈ X possesses a fundamental system of convex neighborhoods. A locally convex topological vector space is a locally convex vector space, whose topology is Hausdorff. Since convexity is translation invariant, for a linear topology T, the local convexity condition (lc) needs only to be verified at x = 0. The following result is a locally convex analogue of Proposition 2.B from TVS I. Proposition 1. In a locally convex space X there exists a basic neighborhood system for 0, which consists of balanced open convex sets. Proof. What we need to prove is that: (∗) for every neighborhood N of 0, there exists a balanced open convex set A ⊂ N . First of all, by definition, there exists a convex neighborhood V of 0, such that V ⊂ N . Secondly, by Proposition 2.B from TVS I, there exists some open balanced set B ⊂ V. Now we are done, by taking A = conv(B). (By CW I, A is balances open convex. Since V is convex and contains B, it will also contain A.) The next result is a locally convex analogue of the Corollary to Theorem 1 from TVS I. It should ne noted, however, that the hypotheses are somehow “more economical” (the collection C could be, for instance, finite), and the conclusion is slightly weaker. Theorem 1. Suppose X is a vector space and C is a collection of balanced1 absorbing convex sets. Then there exists a unique locally convex linear topology T on X , such that the collection U = {εC : C ∈ C, ε > 0} constitutes a fundamental system of T-neighborhoods of 0. Proof. Define the collection V consisting of finite intersections of sets in U. We wish to apply Corollary for Theorem 1 from TVS I, so we need to check the following: 1 The fact that C is balanced forces C 3 0. 1 (i) V is a filter, and all sets in V contain 0; (ii) every V ∈ V is absorbing; (iii) for every V ∈ V, there exists W ∈ V, such that W + W ⊂ V; (iv) every V ∈ V is balanced. Condition (i) is trivial, since all sets in U contain 0. Condition (iv) is also clear, since every set in U is balanced, and (arbitrary) intersections of balanced sets are balanced. Condition (ii) follows from Exercise 2 in CW II, since all sets in U are convex and absorbing. To check condition (iii), start with some V ∈ V, so there exists C1,..., Cn ∈ C and ε1, . , εn > 0, such that V = (ε1C1) ∩ · · · ∩ (εnCn). Consider then the set ε1 εn W = ( 2 C1) ∩ · · · ∩ ( 2 Cn), which again belongs to V. Obviously 2W ⊂ V, and since W is convex, we also have 2W = W + W, so we are done. Having checked the above conditions, we invoke the above mentioned result, to conclude that there exists a unique linear topology T that has V as a basic system of neighborhoods for 0. By construction, T is also the unique linear topology which has U as a fundamental system of neighborhoods for 0. Finally, since all sets in V are convex, it follows that T is indeed locally convex. Remark 1. With C and T as in Theorem 1, the following are equivalent: (i) T is Hausdorff; \ (ii) (εC) = {0}. C∈C ε>0 This is quite clear since, using the notations from the proof, condition (ii) is equivalent to T the condition V∈V V = {0}, which in turn is (see Theorem 1 in TVS I) equivalent to (i). Exercise 1. Suppose that X is locally convex. Prove that the collection of all closed, convex, balanced neighborhoods of 0, constitutes a basis system of neighborhoods of 0. That is, for every neighborhood V of 0, there exists a closed, convex, balanced neighborhood W of 0, such that W ⊂ V. Exercises 2-4. Suppose X is an infinite dimensional vector space. 2. Let C be the collection of all absorbing balanced convex subsets of X , and let T be the corresponding locally convex linear topology described by Theorem 1. Prove that: (i) T is Hausdorff; 2 (ii) if Y is a locally convex vector space, then all linear maps T : X → Y are T- continuous; (iii) all linear subspaces are T -closed; (iv) T is the strongest locally convex linear topology on X . 3. Consider the algebraic dual of X , i.e. the space 0 X = {φ : X → K : φ linear }, 0 0 and define, for φ ∈ C, the set Cφ = {x ∈ X : |φ(x)| < 1}. Let C = {Cφ : φ ∈ X }, and let T0 be the corresponding locally convex linear topology described by Theorem 1. Prove that: (i) T0 is Hausdorff; (ii) all maps φ ∈ X 0 are T’-continuous; (iii) all linear subspaces are T ’-closed. 4. Show that the topology T0 is strictly weaker than T, i.e. there exist T-open sets which are not T0-open. (Hint: Every T0- neighborhood contains an infinite dimensional linear subspace. There are, however, T-neighborhoods of 0 without this property.) p Exercises 5-6 Let 0 < p < 1. We denote by Dp the metric on ` , defined (see TVS I) by ∞ X p p Dp(x, y) = |xn − yn| , ∀ x = (xn), y = (yn) ∈ ` . n=1 For every ρ > 0 we define the ball p Bρ = {x ∈ ` : Dp(x, 0) < ρ}. 5. Show that: sup Dp(x, 0) : x ∈ conv(Bρ) = ∞, ∀ ρ > 0. 6. Use the above Exercise to prove that the metric topology on `p is not locally convex. 3.
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