# Luxemburg norm

A function

$$\| x \| _ {(M)} = \inf \left \{ { \lambda } : {\lambda > 0 , \int\limits _ { G } M ( \lambda ^ {-1} x ( t) ) d t \leq 1 } \right \} ,$$

where $M ( u)$ is an even convex function that increases for positive $u$,

$$\lim\limits _ {u \rightarrow 0 } u ^ {-1} M ( u) = \ \lim\limits _ {u \rightarrow \infty } u ( M ( u) ) ^ {-1} = 0 ,$$

$M ( u) > 0$ for $u > 0$, and $G$ is a bounded set in $\mathbf R ^ {n}$. The properties of this norm were studied by W.A.J. Luxemburg [1]. The Luxemburg norm is equivalent to the Orlicz norm (see Orlicz space), and

$$\| x \| _ {(M)} \leq \| x \| _ {M} \leq 2 \| x \| _ {(M)}.$$

If the functions $M ( u)$ and $N ( u)$ are complementary (or dual) to each other (see Orlicz class), then

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

If $\chi _ {E} ( t)$ is the characteristic function of a measurable subset $E \subset G$, then

$$\| \chi _ {E} \| _ {(M)} = \ \frac{1}{M ^ {-1} ( 1 / \mathop{\rm mes} E ) } .$$

#### References

 [1] W.A.J. Luxemburg, "Banach function spaces" , T.U. Delft (1955) (Thesis) [2] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)
How to Cite This Entry:
Luxemburg norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luxemburg_norm&oldid=55216
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article