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edited May 9, 2018 at 19:22

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The largest topological copy of a Hilbert space contained in $\ell^1$

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asked May 9, 2018 at 18:37

ABB

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The largest copy of a Hilbert space contained in $\ell^1$

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 $\ell^1$?