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

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. 

