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Conjugate class of functions

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A concept in the theory of functions which is a concrete instance of duality in functions spaces. Thus, if a class of functions is regarded as a Banach space or a topological vector space, then the conjugate class is defined as the class of functions isometrically isomorphic to the dual space . For example, when and , there is an isometric isomorphism between the spaces and , under which corresponding elements and are related by

If one considers some class of -periodic summable functions on , then the conjugate class is defined to be the class of functions conjugate to the functions in . For example, the class conjugate to () coincides with the class of functions in for which

The class conjugate to , , coincides with the class of functions in for which .

References

[1] M. Fréchet, C.R. Acad. Sci. , 144 (1907) pp. 1414–1416
[2] F. Riesz, C.R. Acad. Sci. , 144 (1907) pp. 1409–1411
[3] I. [I. Privalov] Priwaloff, Bull. Soc. Math. France , 44 (1916) pp. 100–103
[4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[5] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Conjugate class of functions. T.P. Lukashenko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Conjugate_class_of_functions&oldid=12702
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098