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 ![]() |
[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