ar X iv : m at h / 04 07 20 0 v 2 [ m at h . FA ] 4 A pr 2 00 5 COUNTABLY - NORMED SPACES , THEIR DUAL , AND THE GAUSSIAN MEASURE
نویسنده
چکیده
Here we present an overview of countably-normed spaces. 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. In particular, we show that under certain conditions the strong and inductive topologies coincide and the σ–fields generated by the weak, strong, and inductive topologies are equal. With this σ–field, we develop a Gaussian measure on the dual of a nuclear space. 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 this section we review the basic notions of topological vector spaces along and provide proofs 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.
منابع مشابه
ar X iv : m at h / 04 07 20 0 v 3 [ m at h . FA ] 2 4 A ug 2 00 5 COUNTABLY - NORMED SPACES , THEIR DUAL , AND THE GAUSSIAN MEASURE
Here we present an overview of countably-normed spaces. 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. In particular, we show that under certain conditions the strong and inductive topologies coincide and the σ–fields generated by the weak, strong, and inductive topologies are equa...
متن کاملar 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 ...
متن کاملar X iv : 0 80 4 . 21 32 v 1 [ m at h . D G ] 1 4 A pr 2 00 8 GEOMETRY OF NEUTRAL METRICS IN DIMENSION FOUR ∗
The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces of neutral metrics.
متن کاملar X iv : m at h / 06 12 41 6 v 1 [ m at h . PR ] 1 4 D ec 2 00 6 An L 2 theory for differential forms on path spaces I
An L theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M ...
متن کاملar X iv : m at h / 04 07 22 0 v 1 [ m at h . O A ] 1 3 Ju l 2 00 4 DUALITY AND OPERATOR ALGEBRAS
We investigate some subtle and interesting phenomena in the du-ality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always w *-continuous on dual operator spaces. For example, this yields a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) ...
متن کامل