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)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)
“The new statement will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
—Ralph Waldo Emerson (1803–1882)