Statement
The theorem has several formulations, depending on the context in which it is used. The simplest is sometimes given as follows:
-
- In the plane
- Every continuous function f from a closed disk to itself has at least one fixed point.
This can be generalized to an arbitrary finite dimension:
-
- In Euclidean space
- Every continuous function from a closed ball of a Euclidean space to itself has a fixed point.
A slightly more general version is as follows:
-
- Convex compact set
- Every continuous function f from a convex compact subset K of a Euclidean space to K itself has a fixed point.
An even more general form is better known under a different name:
-
- Schauder fixed point theorem
- Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.
Read more about this topic: Brouwer Fixed-point Theorem
Famous quotes containing the word statement:
“Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
—Robert Benchley (1889–1945)
“He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
—Margaret Halsey (b. 1910)
“No statement about God is simply, literally true. God is far more than can be measured, described, defined in ordinary language, or pinned down to any particular happening.”
—David Jenkins (b. 1925)