On some spaces of vector-valued bounded functions

Abstract

Assuming AE is an algebra of subsets of a non-empty set and X is a normed space, I investigate whether or not certain barrelledness conditions, some of them introduced in the seventies by Saxon and Valdivia, are enjoyed by several subspaces of the linear space of all those bounded X -valued functions defined on which are the uniform limit of a sequence of X-valued AE-simple functions equipped with the supremum-norm. PreliminariesIn what follows will be a non-empty set, AE an algebra of subsets of and X a normed space. By I 0 AEY X we denote the linear space over the field K of the real or complex numbers of all X -valued AE-simple functions defined on . If X K, we will write I 0 AE instead of I 0 AEY K . We represent by B AEY X , or by B Y X , the linear space over K of all bounded X -valued functions defined on which are the uniform limit of a sequence in I 0 AEY X . We will assume that B AEY X is equipped with the supremum-norm f k k sup f 3 k kX 3 P f gand will consider I 0 AEY X as a subspace of B AEY X . If AE is not a '-algebra, then f À1 x , for f P B AEY X and x P X , need not be an element of AE, although certainly it is an element of the '-algebra generated by AE. The subspace of B AEY X formed by all countablyvalued functions will be denoted by K AEY X or by K Y X , while K 0 AEY X will stand for the linear subspace of K AEY X consisting of all those countably-valued f for which there exists a countable partition A n Y n P N f gof by elements of AE such that f is constant in each set A n , n P N. If f P K AEY X is such that f À1 x P AE for all x P X , then clearly f P K 0 AEY X . Naturally K 0 AEY X coincides with K AEY X whenever AE is a '-algebra, although in general K 0 AEY X is a dense subspace of K AEY X , since I 0 AEY X K 0 AEY X . When X K we will write K 0 AE instead of K 0 AEY K . If E we will represent by B EY X the subspace of B AEY X of all those functions f whose