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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.2011-05-26
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    @Theo: I might be slow, but I don't understand your argument. 2. and 3. imply that $c_0$ has subspaces that are not complemented, but to go from there to having subspaces that are not isomorphic to $c_0$, wouldn't you need the converse of 1.?2011-05-26
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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: http://www.springerlink.com/content/7155503p7lx721g4/2011-05-26
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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?2011-05-26
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    @Jonas: The argument I (tried to) give should still be much easier than Gowers's.2011-05-26
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    @Theo: Thanks, I didn't know that.2011-05-26
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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...2011-05-26
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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](http://matematicas.unex.es/~fcabello/files/printable/21.pdf) which looks quite readable at a first glance.2011-05-26
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    @Theo: Thanks very much for the references and additional info. That survey article looks great.2011-05-26

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