FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math » Research
Fixed points of a weakly continuous map f:H->H satisfying f(f(x))=x
Post new topic   Reply to topic Page 1 of 1 [3 Posts] View previous topic :: View next topic
Author Message
Jack Stone
science forum beginner


Joined: 08 Jul 2006
Posts: 3

PostPosted: Mon Jul 10, 2006 3:12 pm    Post subject: Fixed points of a weakly continuous map f:H->H satisfying f(f(x))=x Reply with 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?
Back to top
Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: 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 Reply with quote
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

PostPosted: 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 Reply with quote
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
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [3 Posts] View previous topic :: View next topic
The time now is Thu Feb 21, 2019 5:35 pm | All times are GMT
Forum index » Science and Technology » Math » Research
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts about uni. continuous proof bill1158 Math 1 Tue Jul 18, 2006 10:30 pm
No new posts Tarski fixed-point theorem William Elliot Math 14 Tue Jul 18, 2006 10:24 am
No new posts Absolutely continuous, L^2 question James1118 Math 15 Wed Jul 12, 2006 1:00 pm
No new posts Absolutely continuous, xsin(1/x) James1118 Math 6 Tue Jul 11, 2006 4:54 pm
No new posts Properties of weakly continuous maps in a Hilbert space Jack Stone Math 10 Mon Jul 10, 2006 3:18 pm

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

[ Time: 0.0125s ][ Queries: 16 (0.0023s) ][ GZIP on - Debug on ]