Locally convex topologies on spaces of continuous vector functions

Abstract

Introduction. The strict topology was first defined by BUCK [I] on the space of bounded continuous functions on a locally compact space. SENTILLES [16]

On the spaceC(X, E) with the topology of simple convergence

“…By [11], every tight member of Mp(B(X), E') is >additive and hence it has a unique extension m to a member of M~. v(Bo(X), E') such that the restriction of (m~)p to B(X) coincides with m v For this reason we may consider m as be defined on all of Bo(X).…”

Extension of operators on spaces of vector-valued continuous functions

“…In the topological measure theory the so-called strict topologies on C r c (X, E) are of importance (see [11][12][13][14]19] for definitions and more details). We will now consider the strict topology β σ (X, E) on C r c (X, E).…”

Spaces of Measures as Mackey Completions

“…All locally convex spaces are assumed to be Hausdorff and over K, the field of real or complex numbers. The topologies β 0 β 1 , β, β ∞ , β g , are defined on C b X E in [12,13,22] (see also [5,11,12,15,24]); we also write β σ for β 1 , β τ for β, and β t for β 0 . X ∼ νX will denote the Stone-Čech compactification (real-compactification) of X, θX will denote the topological completion of X, and µX will denote the µ-space associated with X ( [2,24]; note X ⊂ µX ⊂ θX ⊂ νX ⊂ X ∼ .…”