Vol. 106, No. 1, 1983

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Weak compactness in spaces of compact operators and of vector-valued functions

Heron S. Collins and Wolfgang Ruess

Vol. 106 (1983), No. 1, 45–71

This is a study of weak compactness and weak convergence in spaces of compact operators and in spaces of vector-valued functions. The link between these two kinds of spaces is provided by the 𝜀-product X𝜀Y of locally convex spaces X and Y as introduced by Laurent Schwartz. Spaces of compact operators, like K(X,Y ) and X𝜀Y , and spaces of vector-valued functions, like C(K,X), and many more concrete spaces of analysis can be represented as (linear subspaces of) suitable 𝜀-products. Accordingly, the program of this paper is to characterize (i) weak compactness, (ii) weak conditional compactness, (iii) weak sequential convergence, and (iv) reflexivity in the general context of X𝜀Y , and then to specialize the results to (a) spaces of compact operators, (b) injective tensor products, and (c) spaces of vector-valued continuous, or Pettis integrable functions.

Mathematical Subject Classification 2000
Primary: 46M05
Secondary: 46E40, 47D15
Received: 23 September 1981
Published: 1 May 1983
Heron S. Collins
Wolfgang Ruess