Talk:Fixed-point theorems in infinite-dimensional spaces

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)