Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$?

Question

Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$ ?

I only know that any separable Banach space is isometrically isomorphic to a linear subspace of $l^{\infty}$.

Please help . Thanks in advance

The $\ell_p$ spaces for $1\leq p <\infty$ have the property that every infinite dimensional subspace of them contains a further subspace which is isomorphic to the original space. Also $c_0$ shares the same property. Tsirelson (1974) constructed a separable reflexive space which contains no isomorphic copy of any $\ell_p$ space for $1\leq p<\infty$, nor of $c_0$. Combining these two results, you can deduce that Tsirelson's space can't be isomorphically embedded in $c_0$, nor in any $\ell_p$, for $1\leq p<\infty$. — tree detective, Commented Feb 28, 2017 at 20:41
Since you are interested in isometric embeddings, maybe you can find a simpler counterexample. — tree detective, Commented Feb 28, 2017 at 20:50

uniquesolution · Accepted Answer · 2017-02-28 13:09:53Z

2

Of course not. Every closed subspace of $\ell_2$ is isometric to some Hilbert space.

answered Feb 28, 2017 at 13:06

uniquesolution

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Tomasz Kania · Accepted Answer · 2017-03-01 01:23:01Z

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$c_0$ does not embed in $\ell_p$ for $1\leqslant p<\infty$ as it is not weakly sequentially complete.

answered Mar 1, 2017 at 1:23

Tomasz Kania

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2 Answers 2