Completely Continuous Operators
Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence from X, the sequence is norm-convergent in Y (Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact.
Read more about this topic: Compact Operator
Famous quotes containing the words completely and/or continuous:
“One has to completely humiliate oneself to be what the Beatles were, and that’s what I resent. I didn’t know, I didn’t foresee. It happened bit by bit, gradually, until this complete craziness is surrounding you, and you’re doing exactly what you don’t want to do with people you can’t stand—the people you hated when you were ten.”
—John Lennon (1940–1980)
“There was a continuous movement now, from Zone Five to Zone Four. And from Zone Four to Zone Three, and from us, up the pass. There was a lightness, a freshness, and an enquiry and a remaking and an inspiration where there had been only stagnation. And closed frontiers. For this is how we all see it now.”
—Doris Lessing (b. 1919)