Statement
The theorem has several formulations, depending on the context in which it is used. The simplest is sometimes given as follows:
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- 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:
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- 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:
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- 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:
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- 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:
“He that writes to himself writes to an eternal public. That statement only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”
—Ralph Waldo Emerson (1803–1882)
“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)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)