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:
“When I turned into a parent, I experienced a real and total personality change that slowly shifted back to the “normal” me, yet has not completely vanished. I believe the two levels are now superimposed, with an additional sprinkling of mortality intimations.”
—Sonia Taitz (20th century)
“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)