# Conjugate class of functions

A concept in the theory of functions which is a concrete instance of duality in functions spaces. Thus, if a class $ X $
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 $ X ^ {*} $.
For example, when $ 1 \leq p < \infty $
and $ 1 / p + 1 / q = 1 $,
there is an isometric isomorphism between the spaces $ ( L _ {p} [ a , b ] ) ^ {*} $
and $ L _ {q} [ a , b ] $,
under which corresponding elements $ x ^ {*} $
and $ g $
are related by

$$ x ^ {*} ( f ) = \ \int\limits _ { a } ^ { b } g ( x) f ( x) d x . $$

If one considers some class $ X $ of $ 2 \pi $- periodic summable functions on $ [ - \pi , \pi ] $, then the conjugate class is defined to be the class of functions conjugate to the functions in $ X $. For example, the class conjugate to $ L _ {p} [ - \pi , \pi ] $( $ 1 < p < \infty $) coincides with the class of functions $ f $ in $ L _ {p} [ - \pi , \pi ] $ for which

$$ \int\limits _ {- \pi } ^ \pi f ( x) d x = 0 . $$

The class conjugate to $ \mathop{\rm Lip} \alpha $, $ 0 < \alpha < 1 $, coincides with the class of functions in $ \mathop{\rm Lip} \alpha $ for which $ \int _ {- \pi } ^ \pi f ( x) d x = 0 $.

#### 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Conjugate_class_of_functions&oldid=46467