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:
“The atmosphere of orthodoxy is always damaging to prose, and above all it is completely ruinous to the novel, the most anarchical of all forms of literature.”
—George Orwell (1903–1950)
“The habit of common and continuous speech is a symptom of mental deficiency. It proceeds from not knowing what is going on in other people’s minds.”
—Walter Bagehot (1826–1877)