ON THE ULTRABORNOLOGICAL PROPERTY OF THE VECTOR-VALUED BOUNDED-FUNCTION SPACE

If OMEGA is a set, SIGMA a sigma-algebra of subsets of OMEGA and X is a normed space, we show that the space K(SIGMA, X) of all bounded X-countably valued SIGMA-measurable functions on OMEGA endowed with the supremum-norm is ultrabornological if and only if X is ultrabornological. As a consequence, the space l(infinity)(X) of all bounded sequences in X with the supremum-norm is ultrabornological if and only if X is ultrabornological.