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 X⊗𝜀Y , 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.
