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:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“The new statement is always hated by the old, and, to those dwelling in the old, comes like an abyss of skepticism.”
—Ralph Waldo Emerson (1803–1882)
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)