On closed linear subspaces embedded into Banach spaces and their finite-dimensionality

Abstract

This paper studies a Grothendieck-type finite-dimensionality problem for closed linear subspaces embedded in Banach spaces. Let $S_p^{(q)}\subset L_p(M,d\hat{I}¼)$ be a closed linear subspace of the Banach space $L_p(M,d\hat{I}¼)$ defined with respect to a probability measure $d\hat{I}¼$ on $M$. We prove that if $S_p^{(q)}$ is continuously embedded into $L_q(M,d\hat{I}¼)$ for $q>p$, then its dimension $\dim S_p^{(q)}=N\in \mathbb{N}$ satisfies the estimate \[ \frac{1}{N}\left(\frac{\sqrtπ,Γ!\left(\frac{N+\tilde q}{2}\right)}{Γ!\left(\frac{\tilde q+1}{2}\right)Γ!\left(\frac{N}{2}\right)}\right)^{2/\tilde q}\le K_{p,q(m)}^2, \] where $1/\tilde{q}+1/q=1$, $q=2+(p-2)2^m>p$ with $p\ne 2$ and $m\in \mathbb{N}$, and $K_{p,q(m)}>0$ is a bounded constant. We also prove that certain closed linear subspaces of $L_p(M,d\hat{I}¼)$ consisting of continuous functions on $M$ must be finite-dimensional.