0708-1300/the unit sphere in a Hilbert space is contractible
Let H = L2[0,1] and define
For any and any define ft(x) = f for and ft(x) = 1 for . Observe that is continuous and gives the desired retraction to the point f = 1.
This proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
The answer seems to be YES see Spheres in infinite-dimensional normed spaces are Lipschitz contractible