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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> |
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'''Claim''' |
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'''Claim''' |
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'''Proof''' |
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'''Proof''' |
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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.
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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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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? |
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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