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Difference between revisions of "Luxemburg norm"

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$$  
 
$$  
\| x \| _ {(} M)  =  \inf
+
\| x \| _ {(M)} =  \inf
 
\left \{ {
 
\left \{ {
\lambda } : {\lambda > 0 , \int\limits _ { G } M ( \lambda  ^ {-} 1 x ( t) )  d t \leq  1
+
\lambda } : {\lambda > 0 , \int\limits _ { G } M ( \lambda  ^ {-1} x ( t) )  d t \leq  1
 
  } \right \}
 
  } \right \}
 
,
 
,
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$$  
 
$$  
\lim\limits _ {u \rightarrow 0 }  u  ^ {-} 1 M ( u)  = \  
+
\lim\limits _ {u \rightarrow 0 }  u  ^ {-1} M ( u)  = \  
\lim\limits _ {u \rightarrow \infty }  u ( M ( u) )  ^ {-} 1 =  0 ,
+
\lim\limits _ {u \rightarrow \infty }  u ( M ( u) )  ^ {-1}  =  0 ,
 
$$
 
$$
  
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$$  
 
$$  
\| x \| _ {(} M)  \leq  \| x \| _ {M}  \leq  2  \| x \| _ {(} M) .
+
\| x \| _ {(M)} \leq  \| x \| _ {M}  \leq  2  \| x \| _ {(M)}.
 
$$
 
$$
  
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$$  
 
$$  
\| x \| _ {(} M)  = \  
+
\| x \| _ {(M)} = \  
 
\sup  
 
\sup  
 
\left \{ {
 
\left \{ {
\int\limits _ { G } x ( t) y ( t)  d t } : {\| y \| _ {(} N) \leq  1 } \right \}
+
\int\limits _ { G } x ( t) y ( t)  d t } : {\| y \| _ {(N)} \leq  1 } \right \}
 
.
 
.
 
$$
 
$$
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$$  
 
$$  
\| \chi _ {E} \| _ {(} M)  = \  
+
\| \chi _ {E} \| _ {(M)} = \  
  
\frac{1}{M  ^ {-} 1 ( 1 /  \mathop{\rm mes}  E ) }
+
\frac{1}{M  ^ {-1} ( 1 /  \mathop{\rm mes}  E ) }
 
  .
 
  .
 
$$
 
$$

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=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