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