Weak convergence of sequences in function spaces

Abstract

This paper is concerned with weak convergence of sequences of vector valued functions. The functions are to be members of a locally convex space. Thus it is meaningful to speak of the weak topology and weak convergence. Our interest comes from the fact that conditions for the weak convergence of bounded sequences are usually weaker than those for filters and nets. Theorems 224 give such results. A familiar example for scalar valued functions is the space C(K) in which a sequence {LYE} converges weakly to x if and only if {xn> is bounded and converges pointwise to x ([7] p. 265, Cor. 4). We consider a linear space G(X, F) of functions defined on a set X with ranges in a locally convex Hausdorf? space F. Conditions for weak convergence are obtained relative to a given @-topology or G-topology. The results are used to obtain three theorems on weak compactness. One of them, Theorem 6, gives a necessary and sufficient condition and improves on Theorem 17.12 of [9] when restricted to scalar valued functions (see Remark 3). The last section contains an application to the problem of approximating compact operators. When the functions are scalar valued our space is denoted by G(X). In Theorem 1 of [.5j we showed that every locally convex topology