Compact Operator - Completely Continuous Operators

Completely Continuous Operators

Let X and Y be Banach spaces. A bounded linear operator T : XY 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 : XY is compact.

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