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:
“A quality is something capable of being completely embodied. A law never can be embodied in its character as a law except by determining a habit. A quality is how something may or might have been. A law is how an endless future must continue to be.”
—Charles Sanders Peirce (1839–1914)
“If an irreducible distinction between theatre and cinema does exist, it may be this: Theatre is confined to a logical or continuous use of space. Cinema ... has access to an alogical or discontinuous use of space.”
—Susan Sontag (b. 1933)