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:
“Tears are sometimes an inappropriate response to death. When a life has been lived completely honestly, completely successfully, or just completely, the correct response to death’s perfect punctuation mark is a smile.”
—Julie Burchill (b. 1960)
“Perhaps when distant people on other planets pick up some wave-length of ours all they hear is a continuous scream.”
—Iris Murdoch (b. 1919)