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

From Drorbn

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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> |

'''Claim''' | '''Claim''' | ||

Line 7: | Line 7: | ||

'''Proof''' | '''Proof''' | ||

− | + | 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? |

## Revision as of 09:22, 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?