A closed subspace of $c_0$ Does anyone know an example of an infinite dimensional closed linear subspace $S$ of $X=c_0$ (with the sup
norm) which is not isomorphic to $X$, i.e. there does not exist a linear one-to-one map $T$
from $X$ onto $S$ such that both $T$ and its inverse are continuous?
 A: For every sequence $(E_n)$ of finite dimensional Banach spaces and every $\epsilon >0$, there exists a subspace $X$ of $c_0$ that is $(1+\epsilon)$-isomorphic to the $c_0$-sum $(\bigoplus_n E_n)_{c_0}$.
To see this, observe that given a finite-dimensional Banach space $E$ there is $N\in \mathbb{N}$ so large that $E$ $(1+\epsilon)$-embeds into $\ell_\infty^N$ (take $N$ to be the cardinality of a $\delta$-net in the unit ball of $E$, where $\delta$ depends on $\epsilon$). Thus $E$ $(1+\epsilon)$-embeds into $c_0$, and the claim above follows easily.
So it remains to find sequences $(E_n)$ such that $(\bigoplus_n E_n)_{c_0}$ is not isomorphic to $c_0$; such sequences are certainly known. For example, it is known (see Lindenstrauss and Tzafriri's Classical Banach Spaces I, p.73) that $(\bigoplus_n \ell_2^n)_{c_0}$ is not isomorphic to $c_0$. Another example arises by considering $(\bigoplus_n \ell_1^n)_{c_0}$. Using the theory of type/cotype and the theory of [crude] finite representability, one can show that $c_0$ is not crudely finitely representable in $\ell_1 =c_0^*$, whereas $c_0$ is crudely finitely representable in $(\bigoplus_n \ell_1^n)_{c_0}^*$ since $(\bigoplus_n \ell_1^n)_{c_0}^*$ contains uniform copies of $\ell_\infty^n$. Thus $c_0^*$ is not isomorphic to $(\bigoplus_n \ell_1^n)_{c_0}^*$, and so $c_0$ is not isomorphic to $(\bigoplus_n \ell_1^n)_{c_0}$.
