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 closed subspace of $c_0$
-
0@Theo: Thanks very much for the references and additional info. That survey article looks great. – 2011-05-26
1 Answers
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 $(\bigoplus_n \ell_1^n)_{c_0}$.
-
0Thanks for the welcome Theo :) . I might hang here a bit more in future as MO is a bit slow on the Banach spaces front (with some good exceptions, in particular your question about (im)possible spectra for Banach space operators!), and it is good practice to see how other people think about one's own subject area. Of course, I should contribute more Banach space questions to MO than the solitary one that I have asked to date! – 2011-05-31