Trans-Separability in Spaces of Continuous Vector-Valued Functions

Abstract

Abstract. Let X be a completely regular Hausdorff space and E a Hausdorff topological vector space (TVS). In this note, we study the notion of trans-separability for certain subspaces of Cb(X, E) endowed with the c-compact-open and some related topologies.Let X be a completely regular Hausdorff space and E a Hausdorff topological vector space (TVS) with a base W of balanced neighbourhoods of 0, and let Cb(X, E) (resp. C RC (X, E) ) denote the vector space of all continuous i?-valued functions / on X such that F(X) is bounded (resp. relatively compact). Let CQ(X,E) be the subspace of Cb(X,E) consisting of those functions which vanish at infinity.When E is the real or complex field, C B {X,E) = C RC (X,E) and we denote this by Cb(X) and CQ(X, E) by CQ(X). The a-compact-open topology a (resp. compact-open topology k, countable-open topology ao) [4,5,11] on CB(X,E)is defined as the linear topology which has a base of neighbourhoods of 0 consisting of all sets of the form N(A, W) = {fe Cb(X, E) : f(A) C W},where A varies over all er-compact (resp. compact, countable) subsets of X and W € W. The uniform topology u on Cb(X, E) is the linear topology which has a base of neighbourhoods of 0 consisting of all sets of the formwhere W e W. Clearly, a 0 < a < u and k < a. Further, by ([11], Theorem 3.3), a = u iff X = A for some a-compact subset A; k = a iff every cr-compact subset of X is relatively compact; oo = u iff X is separable. For any A C X, let || . m denote the seminorm on Cb(X) given by || / \\ A = sup l6j4 |/(®)|, / e C b (X). We shall denote by C b (X) ® E the vector subspace of Cb(X, E) spaned by the set of all functions of the form 1991 Mathematics Subject Classification: Primary 46E10; Secondary 46E40, 46A10.

Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

“…We mention that, for any neighborhood of 0 in , need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15,16].…”

“…where ⊆ is compact and ∈ W. Clearly, ≤ on ( , ). (For details, see [16].) We state the following two general versions of the Arzelà-Ascoli theorem [16,17] for reference purpose.…”

“…(For details, see [16].) We state the following two general versions of the Arzelà-Ascoli theorem [16,17] for reference purpose. (iii) A vanishes at infinity on ; that is, given ∈ W, there exists a compact set ⊆ such that ( ) ∈ for all ∈ A and ∈ \ ;…”

“…Proof. Since is a quasi-complete TVS, it follows that both ( ( , ), ) and ( (R, ), ) are quasi-complete [13,16,18]. Therefore, precompactness can be considered equivalent to relative compactness in either of these two spaces.…”

On realcompact topological vector spaces

“…The concept of trans-separable spaces has been used to study several problems both from analysis and topology, for example while studying the metrizability of precompact sets in uniform spaces and in the class of lcs, we refer the reader to papers [12,15,21,22,40,41,58,64]. Pfister [57] proved the following:…”