Editing Compact operator (section)

== Completely continuous operators ==
Let ''X'' and ''Y'' be Banach spaces.  A bounded linear operator ''T'' : ''X'' → ''Y'' is called '''completely continuous''' if, for every [[weak topology|weakly convergent]] [[sequence (mathematics)|sequence]] <math>(x_n)</math> from ''X'', the sequence <math>(Tx_n)</math> is norm-convergent in ''Y''  {{harv|Conway|1985|loc=§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.

Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.