0708-1300/the unit sphere in a Hilbert space is contractible

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Let and define


is contractible


For any and any define for and for . Observe that is continuous and gives the desired retraction to the point .

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