Subspaces of \(L_p\) isometric to subspaces of \(\ell_p\)
Three theorems are proven. They are as follows: Theorem A. Let \(0<p<\infty\), \(p\not\in 2\mathbb{N}\). Then a closed linear subspace \(X\) of \(L_p\) is isometrically isomorphic to a subspace of \(\ell_p\) if and only if every, equivalently some, unital subspace which is isometrically isomorphic to \(X\), consists of functions with discrete distribution. Theorem B. Let \(p\in 2\mathbb{N}\). Then every finite-dimensional subspace of \(L_p\) is isometrically isomorphic to a subspace of \(\ell_p\). Moreover, for each \(n\in\mathbb{N}\) there exists an \(N\in\mathbb{N}\), depending on \(n\), \(p\) and \(\mathbb{K}\), such that every \(n\)-dimensional subspace of \(L_p\) is isometrically isomorphic to a subspace of \(\ell^N_p\). Theorem C. Let \(1\leq p<\infty\), \(p\not\in 2\mathbb{N}\), and let \(X\) be a Banach space which is isomorphic either to \(L_p\) or to \(\ell_p\). Then, for each \(k= 2,3,\dots\), there is no finite-dimensional Banach space which is isometrically universal for all \(k\)-dimensional subspaces of \(X\).
scientific article; zbMATH DE number 1228749