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 X and a compact one-one linear operator which maps the conjugate space X' onto a subspace dense in F.It is the purpose of this paper to solve the following problem.Given normed linear space X and Banach space Y, under what conditions does there exist a one-one compact linear map from X onto a subspace dense in F?Necessary and sufficient conditions for such an operator to exist are given (cf.(C) of the theorem below).We first introduce some notations.Suppose X and Fare normed linear spaces.Then ($>(X, Y) (resp., X(X, Y)) is the space of all bounded (resp., compact) linear maps from X to Y. (&0(X, Y) (resp., X,(X, Y)) is the set of all one-one maps in ®(X: Y) (resp., X(X, Y)).®d(X, Y) (resp., Xd(X, Y)) is the set of all maps in (S>(X, Y) (resp., X(X, Y)) with range dense in Y. (&",d(X, Y) = (B0(X, Y)r\(Rd(X, Y), and X,,d(X, Y) = X0(X, Y)C\Xd(X, Y).