Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$.

Convention. For given Hilbert spaces $H$ and $K$, let us write $H\leq K$ if the Hilbertian dimension of $H$ is less than $K$.

Q. What is the Hilbertian dimension of the largest Hilbert space which can be topologically embedded into $\ell^1$?

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    $\begingroup$ Your convention is not used in the question, so why include it? $\endgroup$ – LSpice May 9 '18 at 18:46
  • $\begingroup$ Yes you are right. $\endgroup$ – Ali Bagheri May 9 '18 at 18:49
  • $\begingroup$ relevant: mathoverflow.net/questions/79713/… $\endgroup$ – Pietro Majer May 9 '18 at 19:14
  • $\begingroup$ ANSWER: $\ l^2\ $ is TOPOLOGICALLY embeddable in $\ l^1$. $\endgroup$ – Wlod AA May 10 '18 at 4:10

No infinite dimensional reflexive space can be embedded into $\ell_1$, because every infinite dimensional closed subspace of $\ell_1$ has a non separable dual.

  • $\begingroup$ Since the question is just about a topological embedding, is it obvious that the image is closed? $\endgroup$ – LSpice May 9 '18 at 18:54
  • $\begingroup$ @Pietro Majer What happens if we change the roles? I mean what is the Hilbertian dimension of the smallest Hilbert space contains $\ell^1$ up to topological vector spaces. $\endgroup$ – Ali Bagheri May 9 '18 at 18:55
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    $\begingroup$ @Spice yes because it is complete $\endgroup$ – Pietro Majer May 9 '18 at 18:55
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    $\begingroup$ rmk: Of course if one only wants a homeo onto a subspace, not necessarily linear, then $\ell_2$ is already homeomorphic to $\ell_1$, via $\ell_2\ni x:=(x_j)_j\mapsto (|x_j|x_j)_j\in\ell_1$ $\endgroup$ – Pietro Majer May 9 '18 at 19:03
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    $\begingroup$ @LSpice of course topological embedding (not necessarily linear) need not preserve completeness- e.g. R can be embedded in R as an open interval. But the question is only meaningful in the top. linear sense ($\ell_1$ and $\ell_2$ are already homeomorphic, as topological spaces) $\endgroup$ – Pietro Majer May 9 '18 at 19:37

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