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)