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

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'''Proof'''
 
'''Proof'''
  
Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math>  
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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.
  
Define <math>F_1:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_1(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_2^2}),(1-t)x_2,x_3,x_4,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_2^2}),(1-t)x_2,x_3,x_4,...)||</math>
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Does this proof only works in '''separable''' Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
 
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<math>F_2:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_2(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_3^2}),0,(1-t)x_3,x_4,x_5,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_3^2}),0,(1-t)x_3,x_4,x_5,...)||</math>
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<math>F_3:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_3(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_4^2}),0,0,(1-t)x_4,x_5,x_6,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_4^2}),0,0,(1-t)x_4,x_5,x_6,...)||</math>
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and so on ...
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applying the homotopy <math>F_1</math> in the time interval <math>[0,1/2]</math>, <math>F_2</math> in the interval <math>[1/2,3/4]</math>, <math>F_3</math> in <math>[3/4,5/6]</math> etc...
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we get the desired contraction to the point <math>(1,0,0,...)</math>.
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A different 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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This proof only works in '''separable''' Hilbert spaces. Is the unit ball in a non-separable Hilbert space contractible?
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Revision as of 15:29, 2 November 2007

Let H=\{(x_1,x_2,...)| \sum x_n^2<\infty\} and define S^{\infty}=\{x\in H| ||x||=1\}

Claim

S^{\infty} is contractible

Proof

A way to see this is via the cellular structure of S^{\infty}. If S^{\infty}=C_0 C_1 ... you can always contract C_k along C_{k+1} like moving contracting the equator along the surface of the earth.

Does this proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?