# $K$-convex Banach spaces

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 $$\ell_1^{k_n}$$?

$$Y_n$$ can be $$\ell_\infty^n$$, in which case the best projection onto any $$\ell_1^k$$ is of order at least $$\sqrt{k}$$.
• 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 $n$? May 13, 2021 at 13:07
• 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. May 13, 2021 at 20:42