7.3 Topological Vector Spaces, the Weak and Weak⇤ Topology on Banach Spaces
138 CHAPTER 7. ELEMENTS OF FUNCTIONAL ANALYSIS 7.3 Topological Vector spaces, the weak and weak⇤ topology on Banach spaces The following generalizes normed Vector space. Definition 7.3.1. Let X be a vector space over K, K = C, or K = R, and assume that is a topology on X. we say that X is a topological vector T space (with respect to ), if (X X and K X are endowed with the T ⇥ ⇥ respective product topology) +:X X X, (x, y) x + y is continuous ⇥ ! 7! : K X (λ, x) λ x is continuous. · ⇥ 7! · A topological vector space X is called locally convex,ifeveryx X has a 2 neighborhood basis consisting of convex sets, where a set A X is called ⇢ convex if for all x, y A, and 0 <t<1, it follows that tx +1 t)y A. 2 − 2 In order to define a topology on a vector space E which turns E into a topological vector space we (only) need to define an appropriate neighbor- hood basis of 0. Proposition 7.3.2. Assume that (E, ) is a topological vector space. And T let = U , 0 U . U0 { 2T 2 } Then a) For all x E, x + = x + U : U is a neighborhood basis of x, 2 U0 { 2U0} b) for all U there is a V so that V + V U, 2U0 2U0 ⇢ c) for all U and all R>0 there is a V ,sothat 2U0 2U0 λ K : λ <R V U, { 2 | | }· ⇢ d) for all U and x E there is an ">0,sothatλx U,forall 2U0 2 2 λ K with λ <", 2 | | e) if (E, ) is Hausdor↵, then for every x E, x =0, there is a U T 2 6 2U0 with x U, 62 f) if E is locally convex, then for all U there is a convex V , 2U0 2T with V U.
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