ar X iv : m at h / 04 07 20 0 v 1 [ m at h . FA ] 1 2 Ju l 2 00 4 ABOUT COUNTABLY - NORMED SPACES

Here we present an overview of countably-normed spaces. In particular, we discuss the main topologies—weak, strong, and inductive—placed on the dual of a countably-normed space and discuss the σ-fields generated by these topologies. The purpose in mind is to provide the background material for many of the results used is White Noise Analysis. 1. Topological Vector Spaces In the introduction we place some the basic notions of topological vector spaces along with proofs of a few useful results. 1.1. Topological Preliminaries. Let E be a real vector space. A vector topology τ on E is a topology such that addition E×E → E : (x, y) 7→ x + y and scalar multiplication R × E → E : (t, x) 7→ tx are continuous. If E is a complex vector space we require that C × E → E : (α, x) 7→ αx be continuous. It is useful to observe that when E is equipped with a vector topology, the translation maps tx : E → E : y 7→ y + x are continuous, for every x ∈ E, and are hence also homeomorphisms since t x = t−x. A topological vector space is a vector space equipped with a vector topology. Recall that a local base of a vector topology τ is a family of open sets {Uα}α∈I containing 0 such that if W is any open set containing 0 then W contains some Uα. A set W that contains an open set containing x is called a neighborhood of x. If U is any open set and x any point in U then U − x is an open neighborhood of 0 and hence contains some Uα, and so U itself contains a neighborhood x+ Uα of x: (1.1) If U is open and x ∈ U then x+ Uα ⊂ U , for some α ∈ I Doing this for each point x of U , we see that each open set is the union of translates of the local base sets Uα. If Ux denotes the set of all neighborhoods of a point x in a topological space X , then Ux has the following properties: 1. x ∈ U for all U ∈ Ux 2. if U ∈ Ux and V ∈ Ux, then U ∩ V ∈ Ux 3. if U ∈ Ux and U ⊂ V , then V ∈ Ux. 4. if U ∈ Ux, then there is some V ∈ Ux with U ∈ Uy for all y ∈ V . (taking V to be the interior of U is sufficient). Date: May 13, 2008. 1