ON THE BARRELLEDNESS OF THE VECTOR-VALUED BOUNDED-FUNCTION SPACE

[摘要] If OMEGA is a set, SIGMA a sigma-algebra of subsets of OMEGA, and X a normed space, we show that the space l(infinity)(SIGMA, X) of all bounded X-valued SIGMA-measurable functions defined on OMEGA, provided with the supremum-norm, is barrelled if and only if X is barrelled. Assuming X separable, this implies that the space l(infinity)(OMEGA, X) of all bounded X-valued functions defined on OMEGA, endowed with the supremum-norm, is barrelled whenever X is barrelled. (C) 1994 Academic Press, Inc.

[发布日期] 1994-06-15