Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

Question

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such that given any finite-dimensional subspace $E$ of $X$ there exist a finite-dimensional subspace $F$ of $Y$ with $dim(E)=dim(F)$ and a linear isomorphism $T: E\to{F}$ satisfying $||T||\,||T^{-1}||<\lambda.$ Ribe's Theorem

Ribe, M., On uniformly homeomorphic normed spaces, Ark. Mat. 14, 237-244 (1976). ZBL0336.46018.shows that if $X$ and $Y$ are uniformly homeomorphic Banach spaces, then $X$ is crudely finitely representable in $Y$ and vice versa. Moreover, he proved that

Ribe, M., Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Isr. J. Math. 48, 139-147 (1984). ZBL0577.46015.: let $1<{p_{n},q}<\infty$ and $p_{n}\to{1}$. Then the space $X=(\sum_{n=0}^{\infty}\oplus\ell_{p_{n}})_{q}$ and $\ell_{1}\oplus_{q}X$ are uniformly homeomorphic.

Now, let $(e_{i})$ and $(e_{i}^{n})$ be unit vector basis of $\ell_{1}$ and $\ell_{p_{n}}$, respectively. Set linear span $E_{k}=[e_{1},\cdots,e_{k}]$ and $E_{k,n}=[e_{1}^{n},\cdots, e_{k}^{n}]$. Clearly, $(E_{k})_{k\in\mathbb{N}}$ and $(E_{k,n})_{k\in\mathbb{N}}$ are increasing sequences of subspaces of $\ell_{1}$ and $\ell_{p_{n}}$ such that $\bigcup_{k\in\mathbb{N}}E_{k}$ is dense in $\ell_{1}$ and $\bigcup_{k\in\mathbb{N}}E_{k,n}$ is dense in $\ell_{p_{n}}$, respectively.

Pick $Y_{k}=(E_{k}+E_{k,0}+\cdots+E_{k,k}+0+\cdots)_{q}$ , then $(Y_{k})_{k\in\mathbb{N}}$ are increasing sequences of subspaces of $\ell_{1}\oplus_{q}X$ such that $\bigcup_{k\in\mathbb{N}}Y_{k}$ is dense in $\ell_{1}\oplus_{q}X$.

My question is how to choose finite-dimensional subspaces $(X_{k})_{k\in\mathbb{N}}$ of $X$ so that $\bigcup_{k\in\mathbb{N}}X_{k}$ is dense in $X$ and $dim(X_{k})=dim(Y_{k})$ for each $k\in\mathbb{N}$ and a linear isomorphism $T: X_{k}\to{Y_{k}}$ satisfying $||T||\,||T^{-1}||<\lambda.$ (the constant $\lambda$ is independent of the dimension of finite-dimensional subspaces.) Thank you.

Answer 1

The point is that for every fixed $k$, the Banach-Mazur distance between $E_k \simeq \ell_1^k$ and $E_{k,n}\simeq \ell_{p_n}^k$ goes to $1$ as $n\to \infty$. In particular, there is $n(k) >k$ such that this Banach-Mazur distance is less than $2$ for $n=n(k)$. Then take $X_k=E_{k,n(k)} + E_{k,0}+\dots+E_{k,k}$. The natural map satisfies your condition with $\lambda=2$, but you can take any $\lambda>1$.