Totally bounded sets of precompact linear operators

Abstract

1. Introduction. In this paper several characterizations of totally bounded sets of precompact operators are given. These lead to an affirmative solution of the conjecture that a collectively compact set A is totally bounded iff Si*= {K*: KCSi} is collectively compact. Let 3E and g) be real or complex normed linear spaces with adjoints i* and D*, respectively. Let the closed unit ball in any of these spaces be denoted by attaching a subscript 1, and let [i, g)] be the set of bounded linear operators with domain i and range in $. An operator A£[2E, g)] is called precompact [compact] iff KHi — {Kx: xCXi} is totally bounded [has a compact closure in the norm topology]. If ?) is complete (i.e., a Banach space), then an operator in [i, §)] is precompact iff it is compact, and a subset Si of [x §)] is totally bounded iff its closure is compact. A subset Si of [X, §)] is called collectively compact iff $& = {Kx : K C $ ; x C ii} has compact closure. There is a detailed theory [3] of strongly convergent sequences in a collectively compact set. Characterizing those collectively compact sets which are totally bounded is thus of interest since a strongly (or weakly) convergent sequence converges in norm iff it is contained in a totally bounded set. Investigation of this problem suggested the statements of Theo