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Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic to some closed and bounded subset of H?

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It would suffice to know whether H is homoeomorphic to a closed bounded subset of itself. – Robin Chapman Jun 19 '10 at 19:40
You may want to reprhase the title to actually match the question. – Andrea Ferretti Jun 19 '10 at 22:49
to Robin Chapman: Actually, as you can see from the response below, your question has an affirmative answer-since it has been proved that H is homeomorphic to its own closed unit ball. But I still do not see how I can use this result to prove that my question has an affirmative answer (or that it has a negative answer). – Garabed Gulbenkian Jun 22 '10 at 18:56
I will never understand why it took me so long to see that your question is equivalent to mine but I finally do see it. Thanks for teaching me something about homeomorphisms that I once learned but had somehow forgotten. – Garabed Gulbenkian Jun 28 '10 at 13:47

It is an old result of Klee saying that the infinite-dimensional Hilbert space is homeomorphic with both its unit sphere and its closed unit ball. See e.g. , and the references therein.

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Also, H is homeomorphic with any closed half-space of it, which is the reason why a "C<sup>0</sup> Hilbert manifold with boundary" is in fact a C<sup>0</sup> Hilbert manifold without boundary. – Pietro Majer Jun 19 '10 at 21:46

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