The existence of compact linear maps between Banach spaces

Abstract

In [2] J. D. Weston proves that given any separable Banach space Y, there exist a normed linear space and a compact one-one linear operator which maps the conjugate space X' onto a subspace dense in Y. It is the purpose of this paper to solve the following problem. Given normed linear space and Banach space Y, under what conditions does there exist a one-one compact linear map from onto a subspace dense in Y? Necessary and sufficient conditions for such an operator to exist are given (cf. (C) of the theorem below). We first introduce some notations. Suppose and Y are normed linear spaces. Then 63(X, Y) (resp., ,c(X, Y)) is the space of all bounded (resp., compact) linear maps from to Y. 630(X, Y) (resp., aCO(X, Y)) is the set of all one-one maps in 63(X. Y) (resp., 3C(X, Y)). 63d(X, Y) (resp., aCd(X, Y)) is the set of all maps in 63(X, Y) (resp., 3C(X, Y)) with range dense in Y. 63,d(X, Y) =630(X, Y)n63d(X, Y), and aCo,d(X, Y)=aCo(X, Y)nC\d(X, Y). Finally, 0 is the void set. If is a normed linear space and A is a subset of X', then A is total if and only if for each x $0 in there exists an x' in A such that x'x$0. The following preliminary remarks are easily verified. The first of these gives alternative ways of stating a condition arising prominently in the rest of the paper. (i) If is a normed linear space, then X' contains a countable total subset if and only if X' is separable with respect to the w* topology and also if and only if X' contains a total separable linear subspace. (ii) If is a separable normed linear space, then each of the conjugate spaces X' and X contains a countable total subset. (iii) If and Y are normed linear spaces, 6(30(X, Y) $0, and Y' has a countable total subset, then X' has a countable total subset. DEFINITION. Suppose is a Banach space, and suppose Xk is in and ek is a real number for k=1, 2, * 1. Then {Xk}l iS {ek}lindependent if and only if (a) IIEkXk II < oo and