Convergence of continuous linear functionals and their level sets

Abstract

The origin in both X and X* will be denoted by 0 in the sequel. Corresponding to each y a X* and each c~ a R, we denote the level set {x a X : y(x) = c~} by L(y; e). A natural question to ask is this: can either strong (norm) or weak * convergence of a sequence in X* to a nonzero limit be described in terms of the convergence of the level sets of the linear functionals? We show that when X is complete, weak* convergence of a sequence {Yn: n a/g+} in X* to y 4= 0 is equivalent to the Kuratowski convergence of {L(y,; c0: n e/g + } to L(y; ~) for each real c~. If X is not complete, then weak * convergence does not ensure Kuratowski convergence of level sets. On the other hand, without completeness, strong convergence in X* means convergence of the distance functions for the level sets with respect to the (metrizable) topology of uniform convergence on bounded subsets of X. Most surprisingly, we show that if X is reflexive, then strong convergence in X* is equivalent to the Mosco convergence of level sets. 2. A notational convention. To avoid excessive double subscripting in the sequel, we find it convenient to introduce alternative notation for sequences and their limits (see, e.g., [19]). Suppose N is a cofinal (infinite) subset of the positive integers 2~ +. The notation {x.: n a N} will denote the infinite sequence