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:
“But before the extremity of the Cape had completely sunk, it appeared like a filmy sliver of land lying flat on the ocean, and later still a mere reflection of a sand-bar on the haze above. Its name suggests a homely truth, but it would be more poetic if it described the impression which it makes on the beholder.”
—Henry David Thoreau (1817–1862)
“We read poetry because the poets, like ourselves, have been haunted by the inescapable tyranny of time and death; have suffered the pain of loss, and the more wearing, continuous pain of frustration and failure; and have had moods of unlooked-for release and peace. They have known and watched in themselves and others.”
—Elizabeth Drew (1887–1965)