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)