This is a question I posted on SE, and I have been advised to post it here. 


<https://math.stackexchange.com/questions/146427/subspaces-of-l-p-and-banach-mazur-distance>


It is well-known that every subspace of $l\_2$ is *isometric* to $l\_2$. When $p\neq 2$, $l\_p$ has subspaces that are not even isomorphic, let alone isometric, to $l\_p$. Suppose $X$ is a subspace of $l\_p$ with $p\neq 2$ such that $X$ *is* isomorphic to $l\_p$. What can one say about the Banach-Mazur distance between $X$ and $l\_p$? More precisely, which one of the following mutually exclusive options holds true:


1) Given $K$, there exists a subspace $X$ of $l\_p$, isomorphic to $l\_p$, such that for any isomorphism $T:X\to l\_p$ one has $||T||\cdot||T^{-1}||>K$.


OR


2) There exist a constant $K$ (possibly depending on $p$), such that for any subspace $X$ of $l\_p$, isomorphic to $l\_p$, there exist an isomorphism $T:X\to l\_p$ such that $||T||\cdot||T^{-1}||\leq K$.


Intuitively, I very strongly suspect it is 1) but I do not have an argument to exclude 2) and, if it is indeed 1), I would like to see a concrete example of a subspace having that property.



Subspaces of $l_p$ and Banach-Mazur distance
(1) is correct. It follows from the fact that there is a sequence $(E\_n)$ of finite dimensional subspaces of $\ell\_p$ s.t. $\gamma\_p(E\_n) \to \infty$. Here $\gamma\_p(X)$ is the factorization constant of the identity on $X$ through an $L\_p$ space; that is, $\gamma\_p(X)= \inf \|T\|\cdot \|S\|$, where the infimum is over all $T:X\to Y$, $S:Y:\to X$, and $Y$ an $L\_p$ space. 


For fixed $n$, set $X\_n=\ell\_p(E\_n)$. Since $E\_n$ is finite dimensional, the space $X\_n$ is isomorphic to $\ell\_p$ (and embeds isometrically into $\ell\_p$) but the Banach-Mazur distance of $X\_n$ to $\ell\_p$ is at least $\gamma\_p(E\_n)$.


There remains the sticky point of producing a sequence $(E\_n)$ as above. One way is to take a subspace $X$ of $\ell\_p$ that fails the approximation property and let $E\_n$ be a sequence of finite dimensional subspaces of $X$ with $E\_1\subset E\_2 \subset \dots$ and $\cup E\_n$ dense in $X$. Or, for $p<2$, $E\_n$ can be (Banach-Mazur arbitrarily close to) $\ell\_r^n$ with $p<r<2$, because $\ell\_r$ embeds isometrically into $L\_p(0,1)$ and there are various ways of checking that $\gamma\_p(\ell\_r^n)\to \infty$. Since I do not know your background (you have, unfortunately, chosen to remain anonymous), I do not know which approach of producing $(E\_n)$ is best for you. 


