Vol. 52, No. 2, 1974

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Precompact and collectively semi-precompact sets of semi-precompact continuous linear operators

Andrew S. Geue

Vol. 52 (1974), No. 2, 377–401

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 b[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 b[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 of Z[X,Y ] is said to be collectively semi-precompact if (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 b[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.

Mathematical Subject Classification
Primary: 47D15
Secondary: 46A15
Received: 30 July 1973
Revised: 8 February 1974
Published: 1 June 1974
Andrew S. Geue