A mapping / from a set B into a uniform space (Y,^) is said to be precompact if and only if its range f(B) = {f(b):beB\ is a precompact subset of Y. The precompact subsets of ^^{B, Y), the set of all precompact mappings from B into Y with its natural topology of uniform convergence, are characterized by an Ascoli-Arzela theorem using the notion of equal variation. A linear operator T: X-> Y, where X and Y are topological vector spaces, is said to be semi-precompact if T(B) is precompact for

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... y bounded subset B of X. Let ^[X, Y] denote the set of all continuous linear operators from X into Y with the topology of uniform convergence on bounded subsets of X. Let Jf h [X, Y] denote the subspace of Sf&X, Y] consisting of the semi-precompact continuous linear operators with the induced topology. The precompact subsets of J%^[X, Y] are characterized. A generalized Schauder's theorem for locally convex Hausdorff spaces is obtained. A subset %f of ^[X, Y] is said to be collectively semi-precompact if %f(B) = {H(b):He^beB} is precompact for every bounded subset B of X. Let X and Y be locally convex Hausdorff spaces with Y infrabarrelied. In § 5 the precompact sets of semi-precompact linear operators in Sf h [X 9 Y] are characterized in terms of the concept of collective semi-precompactness of the sets and certain properties of the set of adjoint operators. 1* Introduction* Let X and Y be topological vector spaces over the field of complex numbers C and J*f[X, Y] the set of continuous linear operators from X into Y. For a subset ^f c Jzf[X 9 Y] and a subset B of X, let 3(f(JB) = {H(b): He^beB}. DEFINITION 1.1. A linear operator T: X-* Y is said to be precompact {compact) if there exists a neighborhood V of zero in X such that T(V) is precompact (relatively compact). A linear operator T: X -* Y is said to be semi-precompact (semi-compact) if T(B) is precompact (relatively compact) for every bounded subset B of X. The latter terminology is that of Deshpande and Joshi [14] and coincides with the term "boundedly precompact" used by Ringrose [27] . Clearly, precompactness of an operator is a much stronger 377 378 ANDREW S. GEUE condition than semi-precompactness, unless X has a bounded neighborhood of zero. A precompact operator is always continuous, but this is not the situation for a semi-precompact operator unless we assume X bornological and Y locally convex [18, Proposition l(a), p. 220]. However, we shall always work with continuous semi-precompact operators in this paper and thus avoid the problems otherwise encountered. DEFINITION 1.2. A subset β£*cz^f [X, Y] is said to be collectively precompact (collectively compact) if there exists a neighborhood V of zero in X such that <%?(V) is precompact (relatively compact). A subset 3ff c £f [X, Y] is said to be collectively semi-precompact (collectively semi-compact) if 3ίf(B) is precompact (relatively compact) for every bounded subset B of X.