Abstract:This paper studies a Grothendieck-type finite-dimensionality problem for closed linear subspaces embedded in functional 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 (identically) embedded into $L_q(M,d\mu)$ for $q>p$, then its dimension $\dim S_p^{(q)} = N \in \mathbb{N}$ satisfies the estimate \[ \frac{1}{N}\left(\frac{\sqrt{\pi},\Gamma!\left(\frac{N+\tilde q}{2}\right)}{\Gamma!\left(\frac{\tilde q+1}{2}\right)\Gamma!\left(\frac{N}{2}\right)}\right)^{2/\tilde q}\le K_{p,q(m)}^2, \] where $1/\tilde q + 1/q = 1$, $q = 2 + (p-2)2^m > p$ with $p \neq 2$ 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.
From: Yarema Prykarpatskyy [view email]
[v1]
Wed, 17 Jun 2026 21:21:50 UTC (13 KB)
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