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A function
 
A function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609901.png" /></td> </tr></table>
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$$
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\| x \| _ {(M)}  = \inf
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\left \{ {
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\lambda } : {\lambda > 0 , \int\limits _ { G } M ( \lambda  ^ {-1} x ( t) )  d t \leq  1
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} \right \}
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,
 +
$$
 +
 
 +
where  $  M ( u) $
 +
is an even convex function that increases for positive  $  u $,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609902.png" /> is an even convex function that increases for positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609903.png" />,
+
$$
 +
\lim\limits _ {u \rightarrow 0 }  u  ^ {-1} M ( u)  = \
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\lim\limits _ {u \rightarrow \infty }  u ( M ( u) )  ^ {-1}  = 0 ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609904.png" /></td> </tr></table>
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$  M ( u) > 0 $
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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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609905.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609906.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609907.png" /> is a bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609908.png" />. 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
+
$$
 +
\| x \| _ {(M)}  \leq  \| x \| _ {M}  \leq  2  \| x \| _ {(M)}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l0609909.png" /></td> </tr></table>
+
If the functions  $  M ( u) $
 +
and  $  N ( u) $
 +
are complementary (or dual) to each other (see [[Orlicz class|Orlicz class]]), then
  
If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099011.png" /> are complementary (or dual) to each other (see [[Orlicz class|Orlicz class]]), then
+
$$
 +
\| x \| _ {(M)}  = \
 +
\sup
 +
\left \{ {
 +
\int\limits _ { G } x ( t) y ( t)  d t } : {\| y \| _ {(N)} \leq  1 } \right \}
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099012.png" /></td> </tr></table>
+
If  $  \chi _ {E} ( t) $
 +
is the characteristic function of a measurable subset  $  E \subset  G $,
 +
then
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099013.png" /> is the characteristic function of a measurable subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099014.png" />, then
+
$$
 +
\| \chi _ {E} \| _ {(M)}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060990/l06099015.png" /></td> </tr></table>
+
\frac{1}{M  ^ {-1} ( 1 / \mathop{\rm mes}  E ) }
 +
.
 +
$$
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:39, 19 January 2024


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=17123
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article