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 you model, the focus is completely on you, and some people really appreciate the attention, especially if they didn’t get it growing up. You’re being drawn; you’re being looked at. There’s a sense of acceptance that comes from that.”
—Alexandra Rheault, U.S. model. As quoted in the New York Times, p. 9 (September 6, 1993)
“For good and evil, man is a free creative spirit. This produces the very queer world we live in, a world in continuous creation and therefore continuous change and insecurity.”
—Joyce Cary (1888–1957)