# Definition question: asymptotic-$\ell_{p}$ versus coordinate-free asymptotic-$\ell_{p}$

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 $$y_{i}\in Y_{i}$$.

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.
• 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? Mar 25, 2021 at 19:50
• 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. Mar 26, 2021 at 2:06