0708-1300/the unit sphere in a Hilbert space is contractible: Difference between revisions
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Let <math>H= |
Let <math>H=L^2[0,1]</math> and define <math>S^{\infty}=\{x\in H| ||x||=1\}</math> |
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'''Claim''' |
'''Claim''' |
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'''Proof''' |
'''Proof''' |
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For any <math>t\in[0,1]</math> and any <math>f\in H</math> define <math>f_t(x)= f</math> for <math>0\leq x \leq t</math> and <math>f_t(x)=1</math> for <math>t<x\leq1</math>. |
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Observe that <math>t\rightarrow f_t/||f_t||</math> is continuous and gives the desired retraction to the point <math>f=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? |
Revision as of 10:22, 7 November 2007
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?