0708-1300/the unit sphere in a Hilbert space is contractible: Difference between revisions
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Let <math>H=L^2[0,1]</math> and define <math>S^{\infty}=\{x\in H| ||x||=1\}</math> |
Let <math>H=L^2[0,1]</math> and define <math>S^{\infty}=\{x\in H| ||x||=1\}</math> |
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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] |
Latest revision as of 08:37, 20 November 2007
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Let and define
Claim
is contractible
Proof
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