, Volume 2, Issue 4, pp 339-367

Subspaces of Lp Isometric to Subspaces of ℓ p

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Abstract

We present three results on isometric embeddings of a (closed, linear) subspace X of Lp=Lp[0,1] into ℓ p . First we show that if p ∉ 2N, then X is isometrically isomorphic to a subspace of ℓ p if and only if some, equivalently every, subspace of Lp which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p ∈ 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of ℓ p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of Lp .