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

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