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''completely-continuous mapping''
 
  
A continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239501.png" />, acting from a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239502.png" /> into another space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239503.png" />, that transforms a weakly-convergent sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239504.png" /> to a norm-convergent sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239505.png" />. It is assumed that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239506.png" /> is separable (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239507.png" /> this is not a necessary condition; however, the image of a completely-continuous operator is always separable). In other words, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239508.png" /> is completely-continuous if it maps an arbitrary bounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239509.png" /> into a compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c02395010.png" />. The class of completely-continuous operators is the most important class of the set of compact operators (cf. [[Compact operator|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 [[#References|[1]]] in 1904–1906 for the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c02395011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c02395012.png" /> (cf. [[Hilbert space|Hilbert space]]), by F. Riesz [[#References|[2]]] (the definition in terms of compactness), and, for the general case, by S.S. Banach [[#References|[3]]] (the definition in terms of sequences). The term "compact operatorhas been employed more frequently in recent times, in connection with its appearance in more general topological vector spaces instead of Banach spaces.
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''Completely-Continuous Operator''
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A bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239501.png" />, acting from a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239502.png" /> into another space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239503.png" />, that transforms weakly-convergent sequences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239504.png" /> to norm-convergent sequences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239505.png" />. Equivalently, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239508.png" /> is completely-continuous if it maps every relatively weakly compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239509.png" /> into a relatively compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c02395010.png" />. 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''<sub>1</sub> 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.
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It can be assumed that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239506.png" /> is separable (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023950/c0239507.png" /> this is not a necessary condition; however, the image of a completely-continuous operator is always separable).
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The class of compact operators is the most important class of the set of completely-continuous operators (cf. [[Compact operator|Compact operator]]).
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====References====
 
====References====
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====Comments====
 
====Comments====
The term  "completely-continuous operator"  is in fact not used anymore in Western literature and has been replaced by the term  "compact operator" .
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>

Revision as of 04:14, 31 March 2013


Completely-Continuous Operator

A bounded linear operator , acting from a Banach space into another space , that transforms weakly-convergent sequences in to norm-convergent sequences in . Equivalently, an operator is completely-continuous if it maps every relatively weakly compact subset of into a relatively compact subset of . 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 = l1 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 is separable (for 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)


Comments

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=29567
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article