On the representation of strictly continuous linear functionals

Abstract

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

A Riesz representation theory for completely regular Hausdorff spaces and its applications

“…E 0 of bounded variation .j j.X / < 1/ such that for each x 2 E, x 2 M.X /; where x .A/ WD .A/.x/ for A 2 Bo. Then j j 2 M.X/ (see [8,Lemma 2.3] Definition 2.2. We say that f 2 C b .X; E/ is m-integrable over A 2 Bo provided there exists y A 2 F such that given " > 0 there exists a finite Bo-partition P " of A such that…”

“…Different classes of strict topologiesˇz, where z D ; 1; p; ; t, on the spaces C b .X /; C rc .X; E/; C b .X; E/ are of importance in the topological measure theory and have been studied in numerous papers (see [4][5][6][7][8][9][10][11][12][13][14][15]). Then c ˇt ˇ ˇ1 ˇ u andˇt ˇp ˇ :…”

“…In this paper, we will consider the strict topologyˇt on C b .X; E/ that is also denoted byˇ(see [7,8]) and by o (see [5,11]). It is known that if X is locally compact, then the strictˇt coincides with the original topologyǒ f Buck (see [4]).…”

“…From now on, following [4,7] and [8] we use a symbollˇinstead ofˇt (orˇo) and simply callǎ strict topology.…”