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:
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)