|
Author |
Message |
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 infinite-dimensional 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? |
|
Back to top |
|
 |
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 infinite-dimensional 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 |
|
Back to top |
|
 |
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.ohio-state.edu/~edgar/ |
|
Back to top |
|
 |
Google
|
|
Back to top |
|
 |
|
The time now is Thu Feb 21, 2019 5:35 pm | All times are GMT
|
Copyright © 2004-2005 DeniX Solutions SRL
|
Other DeniX Solutions sites:
Electronics forum |
Medicine forum |
Unix/Linux blog |
Unix/Linux documentation |
Unix/Linux forums |
send newsletters
|
|
Powered by phpBB © 2001, 2005 phpBB Group
|
|