# Completely-continuous operator

*completely-continuous mapping*

A continuous operator , acting from a Banach space into another space , that transforms a weakly-convergent sequence in to a norm-convergent sequence in . It is assumed that the space is separable (for this is not a necessary condition; however, the image of a completely-continuous operator is always separable). In other words, an operator is completely-continuous if it maps an arbitrary bounded subset of into a compact subset of . The class of completely-continuous operators is the most important class of the set of compact operators (cf. Compact operator), which contains, in particular, all compact additive operators.

(Linear) completely-continuous operators were defined and their simplest properties were established by D. Hilbert [1] in 1904–1906 for the spaces and (cf. Hilbert space), by F. Riesz [2] (the definition in terms of compactness), and, for the general case, by S.S. Banach [3] (the definition in terms of sequences). The term "compact operator" has been employed more frequently in recent times, in connection with its appearance in more general topological vector spaces instead of Banach spaces.

#### 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) |

#### Comments

The term "completely-continuous operator" is in fact not used anymore in Western literature and has been replaced by the term "compact operator" .

#### References

[a1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |

[a2] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |

**How to Cite This Entry:**

Completely-continuous operator.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Completely-continuous_operator&oldid=16228