# Difference between revisions of "Completely-continuous operator"

Completely-Continuous Operator

A bounded linear operator \$f\$, acting from a Banach space \$X\$ into another space \$Y\$, that transforms weakly-convergent sequences in \$X\$ to norm-convergent sequences in \$Y\$. Equivalently, an operator \$f\$ is completely-continuous if it maps every relatively weakly compact subset of \$X\$ into a relatively compact subset of \$Y\$. It is easy to see that every compact operator is completely continuous, however the converse is false. For example, recall that the Banach space \$X=l_1\$ has the Schur Property, that is weak sequential and norm sequential convergence coincide. It follows that the identity operator from \$X\$ to \$X\$ is completely-continuous, but it is not compact since \$X\$ is infinite-dimensional. If \$X\$ is reflexive, then every completely-continuous operator is compact, so the two classes of operators do coincide in that case. In the past, the term "completely-continuous operator" was often used to mean compact operator which has sometimes resulted in confusion.

It can be assumed that the space \$X\$ is separable (for \$Y\$ this is not a necessary condition; however, the image of a completely-continuous operator is always separable).

The class of compact operators is the most important class of the set of completely-continuous operators (cf. Compact operator).

#### References

 [1] D. Hilbert, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint (1953) [2] F. Riesz, "Sur les opérations fonctionelles linéaires" C.R. Acad. Sci. Paris Sér. I Math. , 149 (1909) pp. 974–977 [3] S.S. Banach, "Théorie des opérations linéaires" , Hafner (1932)