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Let $(e_{j})_{j=1}^{\infty}$ be a basis for the Banach space $X$. If there exist constants $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$, \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|x_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}x_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|x_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all block sequences $(x_{i})_{i=1}^{N}$ that satisfy $M=M_{N}\leq\min\text{supp}(x_{1})$, then $X$ is said to be (stabilized) asymptotic-$\ell_{p}$ with respect to $(e_{j})_{j=1}^{\infty}$.

There is also coordinate-free generalization of a Banach space being asymptotic-$\ell_{p}$ without reference to a basis. In this situation, there exist $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$, there are subspaces $Y_{1},\ldots,Y_{N}$ of finite-codimension such that \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}y_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all $y_{i}\in Y_{i}$. My question is the following: why do we want the subspaces $Y_{i}$ to have finite codimension? In particular, the block vectors $x_{i}$ in the first definition are members of finite-dimensional subspaces of $X$ (not finite co-dimensional subspaces) and I am wondering why a coordinate-free generalization of the first definition wouldn't take the following form:

$X$ is coordinate-free asymptotic-$\ell_{p}$ if there exist $\zeta_{1},\zeta_{2}>0$ such that for all $N\in\mathbb{N}$ there exist pairwise disjoint finite dimensional subspaces $Y_{1},\ldots,Y_{N}$ of $X$ such that \begin{equation*} \zeta_{1}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}}\leq\left\Vert\sum_{i=1}^{N}y_{i}\right\Vert\leq\zeta_{2}\left(\sum_{i=1}^{N}\|y_{i}\|^{p}\right)^{\frac{1}{p}} \end{equation*} for all $y_{i}\in Y_{i}$.

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Because according to your definition even $\ell_p$, $p\neq 2$ is not asymptotic-$\ell_p$. You can pick your finite dimensional subspaces from a larger Euclidean subspace (by Dvoretsky's theorem). I also don't see how one could salvage your definition.

But you are right that one can define asymptotic structures with respect to different filters. Finite co-dimensional ones, a basis, or even a minimal system can be used. Or you can use weakly null trees, a countably branching tree of finite height where the successor of each node is a weakly null sequence. The basic requirement in all of them is that the finite dimensional spaces you pick are available 'everywhere and arbitrarily spread out in the space'. For reflexive spaces all of the above coincide. The choice of filters are often not terribly important as long as you know what you are after.

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  • $\begingroup$ Thank you for your helpful answer! $\endgroup$
    – JWP_HTX
    Sep 30, 2020 at 21:28
  • $\begingroup$ I certainly agree now that the subspaces $Y_{i}$ must have finite codimension, but I have some related questions. (1) How should I choose the subspaces (tail spaces?) in the first definition so that it also satisfies the correct coordinate-free definition? This is not obvious to me. (2) I suspect that $\ell_{1}(\Gamma)$ for $\Gamma$ uncountable satisfies the coordinate-free definition of being Asymptotic-$\ell_{1}$, but I am having some trouble proving this - is it true? If not, is there a canonical example of a non-separable Asymptotic-$\ell_{1}$ space? $\endgroup$
    – JWP_HTX
    Mar 25, 2021 at 19:50
  • $\begingroup$ Actually, I think I have answered my own questions. The correct coordinate-free notion above glosses over how the subspaces of finite codimension are chosen (i.e. as part of a 2-player game like in Remark 3.8 of this paper by Argyros and Motakis). I would, however, still be interested in examples other than $\ell_{1}(\Gamma)$ of non-separable Asymptotic-$\ell_{1}$ spaces if you happen to know of any. $\endgroup$
    – JWP_HTX
    Mar 26, 2021 at 2:06

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