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

From Drorbn
Revision as of 10:22, 7 November 2007 by Franklin (Talk | contribs)

Jump to: navigation, search

Let H=L^2[0,1] and define S^{\infty}=\{x\in H| ||x||=1\}


S^{\infty} is contractible


For any t\in[0,1] and any f\in H define f_t(x)= f for 0\leq x \leq t and f_t(x)=1 for t<x\leq1. Observe that t\rightarrow f_t/||f_t|| 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?