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Talk:Fixed-point theorems in infinite-dimensional spaces

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This is an old revision of this page, as edited by Linas (talk | contribs) at 22:06, 8 February 2005. The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Linas, you wrote:

Another variant of this theorem states that if U is an open subset of C containing the origin (zero), then any bounded, contractive map f on the closure of U has one, or both of the following properties: (1) f has a unique fixed point, or (2) there is a point x on the boundary of U such that f(x) = a x for some 0 < a < 1.

I don't understand why you need U be a subset of C. It looks to me that you can get away just by requiring U to be open and bounded and not mentioning any C at all. This assuming that your C is the one from the previous paragraph.

So could you please enlighten me. Thanks. Oleg Alexandrov | talk 04:54, 3 Feb 2005 (UTC)

Because the statement applies to the closure of U ... actually, I'm not sure, I scratched my head about this as well. My first impulse is to agree with what you suggest, but I remember puzzling over this as well. The book that I was reading at the time seemed to be careful to make a distinction and also, irritatingly didn't explaining why... I kept that distinction, under the "better safe than sorry" theorem, since I already get into a lot of trouble by going too fast and assuming things that are false... linas 01:16, 8 Feb 2005 (UTC)
Do you happen to still have the book? Note that I removed that text for the time being, since I was not sure of its accuracy and since it can be easily put back if proved correct. Back then I also did not know if you got it from a book or from memory. The reason I see C as unnecessary, is because it can be always manufactured as the convex envelope of U.
If you find your book, I have another question. Is the Schauder fixed point theorem mentioned here the same as the Leray-Shauder theorem (I think I encountered that one a while ago). Thanks. Oleg Alexandrov 01:33, 8 Feb 2005 (UTC)
They're library books, and I have to make a special trip to get to the library :( That, and all the good books are always permanently checked out by someone else anyway ... I'll try to look again. linas 22:06, 8 Feb 2005 (UTC)
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