Embedding nuclear spaces in products of an arbitrary Banach space

Abstract

It is proved that if E is an arbitrary nuclear space and F is an arbitrary infinite-dimensional Banach space, then there exists a fundamental (basic) system of balanced, convex neighborhoods of zero for E such that, for each Vin <, the normed space Ev is isomorphic to a subspace of F. The result for F=l (1 <p< _o) was proved by A. Grothendieck. This paper is an outgrowth of an interest in varieties of topological vector spaces [2] stimulated by J. Diestel and S. Morris, and is in response to their most helpful discussions and questions. The main theorem, valid for arbitrary infinite-dimensional Banach spaces, was first proved by A. Grothendieck [3] (also, see [5, p. 101]) for the Banach spaces I, (1 <p_ oo) and later by J. Diestel for the Banach space c0. Our demonstration relies on two profound results of T. Komura and Y. Komura [4] and C. Bessaga and A. Pelczyn'ski [1], respectively: (i) A locally convex space is nuclear if and only if it is isomorphic to a subspace of a product space (s)', where I is an indexing set and (s) is the Frechet space of all rapidly decreasing sequences. (ii) Every infinite-dimensional Banach space contains a closed infinitedimensional subspace which has a Schauder basis.' Recall that for a balanced, convex neighborhood V of zero in a locally convex space E, EV is a normed space which is norm-isomorphic to (M, pI M), where p is the gauge of V and M is a maximal linear subspace of E on which p is a norm; EV is the completion of Ev. Denote by (s) the nuclear Frechet space of rapidly decreasing sequences, so that (s) = ((n): sup Ink,j < oc, k = 1, 2, 5 Received by the editors February 22, 1971 and, in revised form, October 25, 1971. AMS 1970 subject classifications. Primary 46A05, 46A35, 46B15.