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A mapping f from a set B into
a uniform space (Y,𝒱) is said to be precompact if and only if its range f(B) =
{f(b) : b ∈ B} is a precompact subset of Y . The precompact subsets of 𝒦(B,Y ), the
set of all precompact mappings from B into Y with its natural topology of uniform
convergence, are characterized by an Ascoli-Arzelà 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 every bounded subset B of X.
Let ℒb[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
𝒦y′[X,Y ] denote the subspace of ℒb[X,Y ] consisting of the semi-precompact
continuous linear operators with the induced topology. The precompact subsets of
𝒦b[X,Y ] are characterized. A generalized Schauder’s theorem for locally convex
Hausdorff spaces is obtained. A subset ℋ of Z[X,Y ] is said to be collectively
semi-precompact if ℋ(B) = {H(b) : H ∈ℋ,b ∈ B} is precompact for every
bounded subset B of X. Let X and Y be locally convex Hausdorff spaces
with Y infrabarrelled. In §5 the precompact sets of semi-precompact linear
operators in ℒb[X,Y ] are characterized in terms of the concept of collective
semi-precompactness of the sets and certain properties of the set of adjoint
operators.
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