View source for Luxemburg norm

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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 [[#References|[1]]]. The Luxemburg norm is equivalent to the Orlicz norm (see [[Orlicz space|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.A.J. Luxemburg,   "Banach function spaces" , T.U. Delft  (1955)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   Ya.B. Rutitskii,   "Convex functions and Orlicz spaces" , Noordhoff  (1961)  (Translated from Russian)</TD></TR></table>