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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?

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Not an example but here's an overkill argument why such a space must exist. 1. Every complemented subspace of $c_0$ is isomorphic to $c_0$. 2. $c_{0}$ is not isomorphic to $\ell^2$. 3. A Banach space is isomorphic to a Hilbert space if and only if every closed subspace is complemented. This is not meant very seriously, as the statements of 1. and 3. are very deep theorems. You should be able to find examples or at least pointers to the literature in the book(s) by Lindenstrauss and Tzafriri, Classical Banach Spaces, which unfortunately I don't have access to at the moment. –  t.b. May 26 '11 at 2:58
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Gowers proved that an infinite dimensional Banach space is isomorphic to $\ell^2$ if and only if it is isomorphic to each of its infinite dimensional closed subspaces: springerlink.com/content/7155503p7lx721g4 –  Jonas Meyer May 26 '11 at 3:31
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@Jonas: Very unfortunate formulation on my part and you're absolutely right I wanted to state the converse of 1. (but 1. is also true). 1. Should have been: If $c_{0}$ is contained as a closed subspace of a separable Banach space $E$ then it is complemented in it (and this property characterizes $c_{0}$ up to isomorphism among the separable Banach spaces). According to some notes I have this is discussed in 2.f.5 of LT. This should give the desired conclusion, no? –  t.b. May 26 '11 at 3:40
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@Jonas: Separability of the surrounding space is of course essential, as $c_0$ is not complemented in $\ell^{\infty}$ by Phillips' lemma. By the way, $\ell^{\infty}$ has the same property for all Banach spaces and every Banach space having this subspace implies complemented property is isomorphic to $C(K)$ for $K$ compact and extremally disconnected. E.g. $\ell^{\infty} = C(\beta\mathbb{N})$, but I fear I'm going too far off-topic here, but it is hard for me to resist, as these things were quite crucial for my thesis... –  t.b. May 26 '11 at 4:05
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@Jonas: The result on $c_{0}$ I was referring to is called Sobczyk's theorem. Some digging yielded this survey article which looks quite readable at a first glance. –  t.b. May 26 '11 at 4:55
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1 Answer

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 $(\bigoplus_n \ell_1^n)_{c_0}$.

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Very nice! And a very nice picture of Banach you have there :) –  t.b. May 30 '11 at 16:28
    
Thanks Theo! It is quite a coincidence that I could supply the second example so readily, as the stuff about representability was something I was already in the process of looking into tonight for another reason$*$ when I found this thread (I haven't frequented this site much in the past). –  Philip Brooker May 30 '11 at 17:18
    
$^*$ - on the Ask an Analyst forum, someone asked for example of a sequence of Banach spaces that are isomorphic to $c_0$ but such that the $c_0$-sum of the sequence is not isomorphic to $c_0$; for one of my examples, I used similar arguments to those above involving finite representability. –  Philip Brooker May 30 '11 at 17:21
    
Thanks for that background info! I hope you drop by every once in a while, and share a bit more of your insights on classical Banach space theory, I for one would appreciate it very much. See you around, and welcome to this site, by the way. –  t.b. May 30 '11 at 17:35
    
Thanks 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! –  Philip Brooker May 31 '11 at 5:29
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