Let H be a separable and infinitedimensional Hilbert space. Is every closed subset of H homeomorphic to some closed and bounded subset of H?
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It is an old result of Klee saying that the infinitedimensional Hilbert space is homeomorphic with both its unit sphere and its closed unit ball. See e.g. http://www.ams.org/journals/bull/19616703/S000299041961105892/S000299041961105892.pdf , and the references therein. 

