# Orlicz space

A Banach space of measurable functions introduced by W. Orlicz [1]. Let $ M( u) $
and $ N( u) $
be a pair of complementary $ N $-
functions (see Orlicz class) and let $ G $
be a bounded closed set in $ \mathbf R ^ {n} $.
The Orlicz space $ L _ {M} ^ \star $
is the set of Lebesgue-measurable functions $ x $
on $ G $
for which

$$ \| x \| _ {M} = \sup \left \{ {\int\limits _ { G } x( t) y( t) dt } : { \int\limits _ { G } N( y( t)) dt \leq 1 } \right \} < \infty . $$

An Orlicz space is a complete normed space with respect to the norm $ \| x \| _ {M} $, which is called the Orlicz norm. When $ M( u) = u ^ {p} $, $ 1 < p < \infty $, $ L _ {M} ^ \star $ coincides with the Riesz space $ L _ {p} $ and, up to a scalar factor, $ \| x \| _ {L _ {p} } $ coincides with $ \| x \| _ {M} $.

If $ M _ {1} ( u) $ and $ M _ {2} ( u) $ are $ N $- functions, then the inclusion $ L _ {M _ {1} } ^ \star \subset L _ {M _ {2} } ^ \star $ holds if and only if, for a certain $ C $ and all sufficiently large $ u $, the inequality $ M _ {2} ( u) \leq M _ {1} ( Cu) $ is fulfilled. For every Orlicz space $ L _ {M} ^ \star $ the inclusions $ L _ \infty \subset L _ {M} ^ \star \subset L _ {1} $ hold. Every summable function belongs to some Orlicz space.

The space $ L _ {M} ^ \star $ is separable if and only if $ M( u) $ satisfies the $ \Delta _ {2} $- condition (cf. Orlicz class). In general, $ L _ \infty $ is not dense in $ L _ {M} ^ \star $, and the closure of $ L _ \infty $ in $ L _ {M} ^ \star $ is denoted by $ E _ {M} $ and is always separable. If $ x \in L _ {M} ^ \star $, then

$$ \lim\limits _ {\tau \rightarrow \infty } \sup _ {mes( E) = \tau } \| x {\chi _ {E} } \| _ {M} = \rho ( x, E _ {M} ), $$

where

$$ \chi _ {E} ( t) = \left \{ \begin{array}{ll} 1 , & t \in E, \\ 0, & t \notin E. \\ \end{array} \right .$$

If $ M( u) $ and $ N( u) $ are complementary $ N $- functions and $ x \in L _ {M} ^ \star $, $ y \in L _ {M} ^ \star $, then the following analogue of the Hölder inequality holds:

$$ \int\limits _ { G } x( t) y( t) dt \leq \| x \| _ {(} M) \| y \| _ {(} N) , $$

where $ \| x \| _ {(} M) $ is the Luxemburg norm. Every continuous linear functional $ f $ on $ E _ {M} $ can be represented in the form

$$ f( x) = \int\limits _ { G } x( t) y( t) dt, $$

where $ y \in L _ {N} $ and $ \| f \| = \| y \| _ {(} N) $.

The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces $ L _ {p} $ can also be applied to $ E _ {M} $. The following conditions are equivalent:

1) the space $ L _ {M} ^ \star $ is reflexive;

2) $ M( u) $ and $ N( u) $ satisfy the $ \Delta _ {2} $- condition;

3) an unconditional basis exists in $ L _ {M} ^ \star $;

4) the Haar system forms an unconditional basis in $ L _ {M} ^ \star $;

5) the trigonometric system is a basis in $ L _ {M} ^ \star $ and the Haar system is a basis in $ E _ {M} $.

A sequence space $ l _ {M} ^ \star $ is defined in the same way, but the properties of $ l _ {M} ^ \star $ depend on the asymptotics of the function $ M( u) $ at 0. Many of the geometric properties of $ L _ {M} ^ \star $ and $ l _ {M} ^ \star $ are studied in [5]; for example, for any function $ M( u) $, the set of all $ p $ such that $ l _ {p} $ is isomorphically imbeddable in $ L _ {M} ^ \star $ 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) |

**How to Cite This Entry:**

Orlicz space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Orlicz_space&oldid=49503