A Banach space of measurable functions introduced by W. Orlicz . Let and be a pair of complementary -functions (see Orlicz class) and let be a bounded closed set in . The Orlicz space is the set of Lebesgue-measurable functions on for which
An Orlicz space is a complete normed space with respect to the norm , which is called the Orlicz norm. When , , coincides with the Riesz space and, up to a scalar factor, coincides with .
If and are -functions, then the inclusion holds if and only if, for a certain and all sufficiently large , the inequality is fulfilled. For every Orlicz space the inclusions hold. Every summable function belongs to some Orlicz space.
The space is separable if and only if satisfies the -condition (cf. Orlicz class). In general, is not dense in , and the closure of in is denoted by and is always separable. If , then
If and are complementary -functions and , , then the following analogue of the Hölder inequality holds:
where is the Luxemburg norm. Every continuous linear functional on can be represented in the form
where and .
The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces can also be applied to . The following conditions are equivalent:
1) the space is reflexive;
2) and satisfy the -condition;
3) an unconditional basis exists in ;
4) the Haar system forms an unconditional basis in ;
5) the trigonometric system is a basis in and the Haar system is a basis in .
A sequence space is defined in the same way, but the properties of depend on the asymptotics of the function at 0. Many of the geometric properties of and are studied in ; for example, for any function , the set of all such that is isomorphically imbeddable in can be found.
Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.
|||W. Orlicz, "Ueber eine gewisse Klasse von Räumen vom Typus " Bull. Intern. Acad. Pol. Ser. A , 8/9 (1932) pp. 207–220|
|||M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)|
|||V.F. Gaposhkin, "Existence of absolute bases in Orlicz spaces" Funct. Anal. Appl. , 1 : 4 (1967) pp. 278–284 Funkts. Anal. i Prilozhen. , 1 : 4 (1967) pp. 26–32|
|||S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)|
|||J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)|
|[a1]||W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)|
Orlicz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz_space&oldid=17861