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

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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>
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'''


Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math>
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>.
Observe that <math>t\rightarrow f_t/||f_t||</math> is continuous and gives the desired retraction to the point <math>f=1</math>.


This proof only works in '''separable''' Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?
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>


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]
<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>

<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>

and so on ...

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...

we get the desired contraction to the point <math>(1,0,0,...)</math>.

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

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