0708-1300/the unit sphere in a Hilbert space is contractible
Let and define
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?