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Let and define
Claim
is contractible
Proof
Suppose then
Define by
by
by
and so on ...
applying the homotopy in the time interval , in the interval , in etc...
we get the desired contraction to the point .
A different way to see this is via the cellular structure of . If you can always contract along like moving contracting the equator along the surface of the earth.
This proof only works in separable Hilbert spaces. Is the unit ball in a non-separable Hilbert space contractible?