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:
“You know that fiction, prose rather, is possibly the roughest trade of all in writing. You do not have the reference, the old important reference. You have the sheet of blank paper, the pencil, and the obligation to invent truer than things can be true. You have to take what is not palpable and make it completely palpable and also have it seem normal and so that it can become a part of experience of the person who reads it.”
—Ernest Hemingway (1899–1961)
“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)