Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$. A famous theorem of Pisier's asserts that an infinite dimensional Banach space is $K$-convex iff it does not contain $\ell_1^n$'s uniformly. For the notion of $K$-convexity look at https://bookstore.ams.org/cbms-60. I want to know the following suppose an infinite dimensional Banach space $X$ is not $K$-convex. Now assume that $X$ is an inductive limit of Banach spaces $Y_1\subseteq Y_2\subseteq\dots\subseteq Y_n\subseteq\dots$ where $\text{dim}Y_n=n.$ Let $X$ not $K$-convex. Clearly without loss of generality we may assume that there exists an increasing sequence $(k_n)$ such that $Y_n$ contains a subspace of dimension $k_n$ which is $\lambda$-isomorphic to $\ell_1^{k_n}$ for all $n\geq 1$ and $\lambda\geq 1$ fixed positive constant. My question is if there are any known estimates on the norm of projection from $Y_n$ to the subspace which is $\lambda$-isomorphic to $\ell_1^{k_n}$?
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asked May 12, 2021 at 9:44
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$Y_n$ can be $\ell_\infty^n$, in which case the best projection onto any $\ell_1^k$ is of order at least $\sqrt{k}$.
answered May 12, 2021 at 17:49
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$\begingroup$ I am starting with the Banach spaces $Y_n$'s and now I want to know a bound on the projection constant. For example $Y_n=\ell_p^n$ and $X=\ell_p.$ Or do you want to say that if X is not K-convex then X contains $Y_n$'s isomorphic to $\ell_\infty^n$'s for which you have what you are saying? If so can you also find the bounds on norms of the isomorphism depending on $n$? $\endgroup$ May 13, 2021 at 13:07
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$\begingroup$ Also please note that I have the restriction that X is not K convex. So there might be something special happening. $\endgroup$ May 13, 2021 at 14:06
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2$\begingroup$ If the $Y_n$ are uniformly isomorphic to $\ell_n^p$, with $1<p<\infty$, then $X$ is $K$-convex. My answer implies that in general you do not have bounds better than of order $\sqrt{n}$ if, e.g., $X$ is a $C(K)$ space. $\endgroup$ May 13, 2021 at 20:42