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

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Let <math>H=\{(x_1,x_2,...)| \sum x_n^2<\infty\}</math> and define <math>S^{\infty}=\{x\in H| ||x||=1\}</math>
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Let <math>H=L^2[0,1]</math> and define <math>S^{\infty}=\{x\in H| ||x||=1\}</math>
  
 
'''Claim'''
 
'''Claim'''
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'''Proof'''
 
'''Proof'''
  
A way to see this is via the cellular structure of <math>S^{\infty}</math>. If <math>S^{\infty}=C_0 C_1 ...</math> you can always contract <math>C_k</math> along <math>C_{k+1}</math> like moving contracting the equator along the surface of the earth.
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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>.
  
Does this proof only works in '''separable''' Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
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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 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?