Structure of Local Banach Spaces of Locally Convex Spaces

1. Introduction and notation. In this paper the word " local " is used in at least three different meanings. Our aim is to study local Banach spaces of Fréchet or other locally convex spaces, and it turns out that it is convenient to use the local theory of Banach spaces for this purpose. Recall that given a locally convex space E and a continuous seminorm p on E the completion of the normed space (E/ker(p), p) is a local Banach space of E, and it is denoted by E p. If (p α) α∈A is a system of seminorms defining the topology of E, then we call (E p α) α∈A a system of local Banach spaces. Valdivia proved in [V] that every infinite dimensional nuclear locally convex space E has for all separable infinite dimensional Banach spaces X a system of local Banach spaces isomorphic to X. In this paper we study Schwartz spaces and, consequently, compact linear operators in Banach spaces. We consider " dense " factorizations of operators in Banach spaces, i.e., factorizations T = T (2) T (1) , T ∈ L(X, Y), through Z such that T (1) (X) ⊂ Z is dense. Assuming density is enough to guarantee a direct application to the study of local Banach spaces of locally convex spaces. In Section 2. we introduce a way to measure the distance of nonisomorphic Banach spaces, the so called local distance function. We prove general theorems (Theorems 2.8 and 2.14) which show that a compact operator T on a separable reflexive Banach space X factors through a Banach space Y in a strong sense (the image of X is dense in Y ; both factors of the given operator are compact), if the distance of X * and Y * is small enough and some technical assumptions are satisfied. For example we show that an arbitrary compact operator on a separable L p –space X, 1 < p < ∞, factors through any separable L p –space Y in the above strong sense (Theorem 2.10). Consequently, if X and Y are such spaces, we see that a Schwartz space having a system of local Banach space isomorphic to X also has a system isomorphic to Y (Corollary 2.16). The concept of local distance function is also analyzed from the Banach space theoretical point of view. We show (Proposition 2.4) that there exist …