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

From Drorbn
Jump to navigationJump to search
No edit summary
No edit summary
Line 1: Line 1:
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'''
Line 7: Line 7:
'''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.
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>.


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

Retrieved from "https://drorbn.net/index.php?title=0708-1300/the_unit_sphere_in_a_Hilbert_space_is_contractible&oldid=5865"