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Jack Stone science forum beginner
Joined: 08 Jul 2006
Posts: 3

Posted: Mon Jul 10, 2006 3:12 pm Post subject:
Fixed points of a weakly continuous map f:H>H satisfying f(f(x))=x



Let H be an infinitedimensional Hilbert space.
Does there exist a mapping f:H>H so that:
i) f is weakly continuous,
ii) f(f(x))=x for all x in H,
iii) f has no fixed points (for no x f(x)=x)
or
every mapping f:H>H satisfying i) and ii) must have a fixed point? 

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Lee Rudolph science forum Guru
Joined: 28 Apr 2005
Posts: 566

Posted: Mon Jul 10, 2006 9:19 pm Post subject:
Re: Fixed points of a weakly continuous map f:H>H satisfying f(f(x))=x



Jack Stone <stone1292@mail.com> writes:
Quote:  Let H be an infinitedimensional Hilbert space.
Does there exist a mapping f:H>H so that:
i) f is weakly continuous,
ii) f(f(x))=x for all x in H,
iii) f has no fixed points (for no x f(x)=x)
or
every mapping f:H>H satisfying i) and ii) must have a fixed point?

Let h:H>S be a diffeomorphism from H to the unit sphere S in H;
it's known that such an h exists. Let f = h^{1} o A o h, where
A is the antipodal map on S, and there you are.
Lee Rudolph 

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G. A. Edgar science forum Guru
Joined: 29 Apr 2005
Posts: 470

Posted: Tue Jul 11, 2006 12:47 pm Post subject:
Re: Fixed points of a weakly continuous map f:H>H satisfying f(f(x))=x



Quote: 
Let h:H>S be a diffeomorphism from H to the unit sphere S in H;
it's known that such an h exists.

But not weakly continuous...
The closed unit ball of H is weakly compact and convex, so every weakly
continuous map has a fixed point. This doesn't solve the
original question, but does imply that f weakly continuous with no
fixed points cannot map any ball into itself.

G. A. Edgar http://www.math.ohiostate.edu/~edgar/ 

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