On the embedding of λ-nuclear spaces into product of Banach spaces

Abstract

If E is the nuclear space s of rapidly decreasing sequences and F is any infinitedimensional Banach space with Schauder basis, then for every 0-neighborhood U in E there exists an absolutely convex 0-neighborhood Vc U in E such that /~v is norm-isomorphic to F (see [6]). This result of Saxon was improved later on by Valdivia [8] proving its validity when E is an arbitrary nuclear space and F any infinite-dimensional separable Banach space. In the present paper we bring up these results into the context of 2-nuclearity. Namely, we will prove that for certain sequence spaces 4, a Mackey space a can be found satisfying the following condition: If F is any infinite-dimensional Banach space with Schauder basis, for every 0-neighborhood U in othere is an absolutely convex 0-neighborhood V in osuch that ffv is norm-isomorphic to F. As a consequence we prove an embedding theorem of k-nuclear spaces into some product of any given infinite-dimensional Banach space with Schauder basis. This research, supported by the University of Extremadura, was carried out during the author's visit to the University of Kaiserslautern (F.R.G.).