Statement
Kakutani's theorem states:
- Let S be a non-empty, compact and convex subset of some Euclidean space Rn. Let φ: S → 2S be a set-valued function on S with a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then φ has a fixed point.
When we say that the graph of is closed, we mean that for all sequences and such that, and for all, we have .
Read more about this topic: Kakutani Fixed-point Theorem
Famous quotes containing the word statement:
“The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.”
—Polly Berrien Berends (20th century)
“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)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)