Orlicz space
A Banach space of measurable functions introduced by W. Orlicz [1]. 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
 |
where
 |
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 [5]; 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.
References
| [1] | W. Orlicz, "Ueber eine gewisse Klasse von Räumen vom Typus  " Bull. Intern. Acad. Pol. Ser. A , 8/9 (1932) pp. 207–220 |
| [2] | M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian) |
| [3] | 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 |
| [4] | S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian) |
| [5] | J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979) |
Comments
References
| [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=48069