The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if is a convex subset of a topological vector space and is a continuous mapping of into itself so that is contained in a compact subset of, then has a fixed point.
A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was discovered earlier by Juliusz Schauder and Jean Leray. The statement is as follows:
Let be a continuous and compact mapping of a Banach space into itself, such that the set
is bounded. Then has a fixed point.
Read more about Schauder Fixed Point Theorem: History
Famous quotes containing the words fixed, point and/or theorem:
“Marriage is the clue to human life, but there is no marriage apart from the wheeling sun and the nodding earth, from the straying of the planets and the magnificence of the fixed stars.”
—D.H. (David Herbert)
“Our culture still holds mothers almost exclusively responsible when things go wrong with the kids. Sensing this ultimate accountability, women are understandably reluctant to give up control or veto power. If the finger of blame was eventually going to point in your direction, wouldn’t you be?”
—Ron Taffel (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)