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:
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in ones theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (18921983)
“After the first powerful plain manifesto
The black statement of pistons, without more fuss
But gliding like a queen, she leaves the station.”
—Stephen Spender (19091995)
“I think, therefore I am is the statement of an intellectual who underrates toothaches.”
—Milan Kundera (b. 1929)