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

From Drorbn

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?