Embedding Nuclear Spaces in Products of an Arbitrary Banach Space

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 <!-- MATH $\mathcal{V}$ --> of balanced, convex neighborhoods of zero for E such that, for each V in <!-- MATH $\mathcal{V}$ --> , the normed space is isomorphic to a subspace of F. The result for <!-- MATH $F = {l_p}\;(1 \leqq p \leqq \infty )$ --> was proved by A. Grothendieck.