Difference between revisions of "0708-1300/the unit sphere in a Hilbert space is contractible"

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This proof only works in '''separable''' Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
 
This proof only works in '''separable''' Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
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The answer seems to be '''YES''' see [http://www.jstor.org/view/00029939/di970909/97p01032/0?frame=noframe&userID=80644483@utoronto.ca/01c0a80a6600501ced693&dpi=3&config=jstor Spheres in infinite-dimensional normed spaces are Lipschitz contractible]

Revision as of 10:44, 7 November 2007

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

Claim

S^{\infty} is contractible

Proof

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?

The answer seems to be YES see Spheres in infinite-dimensional normed spaces are Lipschitz contractible