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A [[Banach space|Banach space]] of measurable functions introduced by W. Orlicz [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702202.png" /> be a pair of complementary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702204.png" />-functions (see [[Orlicz class|Orlicz class]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702205.png" /> be a bounded closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702206.png" />. The Orlicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702207.png" /> is the set of Lebesgue-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702208.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o0702209.png" /> for which
+
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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/o/o070/o070220/o07022010.png" /></td> </tr></table>
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An Orlicz space is a complete normed space with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022011.png" />, which is called the Orlicz norm. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022014.png" /> coincides with the [[Riesz space|Riesz space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022015.png" /> and, up to a scalar factor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022016.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022017.png" />.
+
A [[Banach space|Banach space]] of measurable functions introduced by W. Orlicz [[#References|[1]]]. Let  $  M( u) $
 +
and  $  N( u) $
 +
be a pair of complementary  $  N $-
 +
functions (see [[Orlicz class|Orlicz class]]) and let  $  G $
 +
be a bounded closed set in  $  \mathbf R  ^ {n} $.  
 +
The Orlicz space  $  L _ {M}  ^  \star  $
 +
is the set of Lebesgue-measurable functions  $  x $
 +
on  $  G $
 +
for which
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022019.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022021.png" />-functions, then the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022022.png" /> holds if and only if, for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022023.png" /> and all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022024.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022025.png" /> is fulfilled. For every Orlicz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022026.png" /> the inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022027.png" /> hold. Every summable function belongs to some Orlicz space.
+
$$
 +
\| x \| _ {M}  = \sup \left \{ {\int\limits _ { G } x( t) y( t)  dt } : {
 +
\int\limits _ { G } N( y( t))  dt \leq  1 } \right \}
 +
< \infty .
 +
$$
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022028.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022029.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022031.png" />-condition (cf. [[Orlicz class|Orlicz class]]). In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022032.png" /> is not dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022033.png" />, and the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022035.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022036.png" /> and is always separable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022037.png" />, then
+
An Orlicz space is a complete normed space with respect to the norm  $  \| x \| _ {M} $,
 +
which is called the Orlicz norm. When  $  M( u) = u  ^ {p} $,
 +
$  1 < p < \infty $,
 +
$  L _ {M}  ^  \star  $
 +
coincides with the [[Riesz space|Riesz space]] $  L _ {p} $
 +
and, up to a scalar factor, $  \| x \| _ {L _ {p}  } $
 +
coincides with  $  \| 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/o/o070/o070220/o07022038.png" /></td> </tr></table>
+
If  $  M _ {1} ( u) $
 +
and  $  M _ {2} ( u) $
 +
are  $  N $-
 +
functions, then the inclusion  $  L _ {M _ {1}  }  ^  \star  \subset  L _ {M _ {2}  }  ^  \star  $
 +
holds if and only if, for a certain  $  C $
 +
and all sufficiently large  $  u $,
 +
the inequality  $  M _ {2} ( u) \leq  M _ {1} ( Cu) $
 +
is fulfilled. For every Orlicz space  $  L _ {M}  ^  \star  $
 +
the inclusions  $  L _  \infty  \subset  L _ {M}  ^  \star  \subset  L _ {1} $
 +
hold. Every summable function belongs to some Orlicz space.
 +
 
 +
The space  $  L _ {M}  ^  \star  $
 +
is separable if and only if  $  M( u) $
 +
satisfies the  $  \Delta _ {2} $-
 +
condition (cf. [[Orlicz class|Orlicz class]]). In general,  $  L _  \infty  $
 +
is not dense in  $  L _ {M}  ^  \star  $,
 +
and the closure of  $  L _  \infty  $
 +
in  $  L _ {M}  ^  \star  $
 +
is denoted by  $  E _ {M} $
 +
and is always separable. If  $  x \in L _ {M}  ^  \star  $,
 +
then
 +
 
 +
$$
 +
\lim\limits _ {\tau \rightarrow \infty }  \sup _ {mes( E) = \tau }  \| x {\chi _ {E} } \| _ {M}  =  \rho ( x, E _ {M} ),
 +
$$
  
 
where
 
where
  
<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/o/o070/o070220/o07022039.png" /></td> </tr></table>
+
$$
 +
\chi _ {E} ( t)  = \left \{
 +
 
 +
\begin{array}{ll}
 +
1 ,  & t \in E,  \\
 +
0,  & t \notin E. \\
 +
\end{array}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022041.png" /> are complementary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022042.png" />-functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022044.png" />, then the following analogue of the [[Hölder inequality|Hölder inequality]] holds:
+
\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/o/o070/o070220/o07022045.png" /></td> </tr></table>
+
If  $  M( u) $
 +
and  $  N( u) $
 +
are complementary  $  N $-
 +
functions and  $  x \in L _ {M}  ^  \star  $,
 +
$  y \in L _ {M}  ^  \star  $,
 +
then the following analogue of the [[Hölder inequality|Hölder inequality]] holds:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022046.png" /> is the [[Luxemburg norm|Luxemburg norm]]. Every continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022048.png" /> can be represented in the form
+
$$
 +
\int\limits _ { G } x( t) y( t)  dt  \leq  \| x \| _ {(} M) \| y \| _ {(} N) ,
 +
$$
  
<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/o/o070/o070220/o07022049.png" /></td> </tr></table>
+
where  $  \| x \| _ {(} M) $
 +
is the [[Luxemburg norm|Luxemburg norm]]. Every continuous linear functional  $  f $
 +
on  $  E _ {M} $
 +
can be represented in the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022051.png" />.
+
$$
 +
f( x)  = \int\limits _ { G } x( t) y( t)  dt,
 +
$$
  
The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022052.png" /> can also be applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022053.png" />. The following conditions are equivalent:
+
where  $  y \in L _ {N} $
 +
and $  \| f \| = \| y \| _ {(} N) $.
  
1) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022054.png" /> is reflexive;
+
The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces  $  L _ {p} $
 +
can also be applied to  $  E _ {M} $.
 +
The following conditions are equivalent:
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022056.png" /> satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022057.png" />-condition;
+
1) the space  $  L _ {M}  ^  \star  $
 +
is reflexive;
  
3) an unconditional [[Basis|basis]] exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022058.png" />;
+
2) $  M( u) $
 +
and  $  N( u) $
 +
satisfy the  $  \Delta _ {2} $-
 +
condition;
  
4) the [[Haar system|Haar system]] forms an unconditional basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022059.png" />;
+
3) an unconditional [[Basis|basis]] exists in $  L _ {M}  ^  \star  $;
  
5) the trigonometric system is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022060.png" /> and the Haar system is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022061.png" />.
+
4) the [[Haar system|Haar system]] forms an unconditional basis in $  L _ {M}  ^  \star  $;
  
A sequence space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022062.png" /> is defined in the same way, but the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022063.png" /> depend on the asymptotics of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022064.png" /> at 0. Many of the geometric properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022066.png" /> are studied in [[#References|[5]]]; for example, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022067.png" />, the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022069.png" /> is isomorphically imbeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022070.png" /> can be found.
+
5) the trigonometric system is a basis in  $  L _ {M}  ^  \star  $
 +
and the Haar system is a basis in  $  E _ {M} $.
 +
 
 +
A sequence space $  l _ {M}  ^  \star  $
 +
is defined in the same way, but the properties of $  l _ {M}  ^  \star  $
 +
depend on the asymptotics of the function $  M( u) $
 +
at 0. Many of the geometric properties of $  L _ {M}  ^  \star  $
 +
and $  l _ {M}  ^  \star  $
 +
are studied in [[#References|[5]]]; for example, for any function $  M( u) $,  
 +
the set of all $  p $
 +
such that $  l _ {p} $
 +
is isomorphically imbeddable in $  L _ {M}  ^  \star  $
 +
can be found.
  
 
Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.
 
Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.
Line 43: Line 130:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Orlicz,  "Ueber eine gewisse Klasse von Räumen vom Typus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022071.png" />"  ''Bull. Intern. Acad. Pol. Ser. A'' , '''8/9'''  (1932)  pp. 207–220</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><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Existence of absolute bases in Orlicz spaces"  ''Funct. Anal. Appl.'' , '''1''' :  4  (1967)  pp. 278–284  ''Funkts. Anal. i Prilozhen.'' , '''1''' :  4  (1967)  pp. 26–32</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1–2''' , Springer  (1977–1979)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Orlicz,  "Ueber eine gewisse Klasse von Räumen vom Typus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070220/o07022071.png" />"  ''Bull. Intern. Acad. Pol. Ser. A'' , '''8/9'''  (1932)  pp. 207–220</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><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Gaposhkin,  "Existence of absolute bases in Orlicz spaces"  ''Funct. Anal. Appl.'' , '''1''' :  4  (1967)  pp. 278–284  ''Funkts. Anal. i Prilozhen.'' , '''1''' :  4  (1967)  pp. 26–32</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.G. Krein,  Yu.I. Petunin,  E.M. Semenov,  "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Lindenstrauss,  L. Tzafriri,  "Classical Banach spaces" , '''1–2''' , Springer  (1977–1979)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>

Latest revision as of 14:54, 7 June 2020


A Banach space of measurable functions introduced by W. Orlicz [1]. Let $ M( u) $ and $ N( u) $ be a pair of complementary $ N $- functions (see Orlicz class) and let $ G $ be a bounded closed set in $ \mathbf R ^ {n} $. The Orlicz space $ L _ {M} ^ \star $ is the set of Lebesgue-measurable functions $ x $ on $ G $ for which

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

An Orlicz space is a complete normed space with respect to the norm $ \| x \| _ {M} $, which is called the Orlicz norm. When $ M( u) = u ^ {p} $, $ 1 < p < \infty $, $ L _ {M} ^ \star $ coincides with the Riesz space $ L _ {p} $ and, up to a scalar factor, $ \| x \| _ {L _ {p} } $ coincides with $ \| x \| _ {M} $.

If $ M _ {1} ( u) $ and $ M _ {2} ( u) $ are $ N $- functions, then the inclusion $ L _ {M _ {1} } ^ \star \subset L _ {M _ {2} } ^ \star $ holds if and only if, for a certain $ C $ and all sufficiently large $ u $, the inequality $ M _ {2} ( u) \leq M _ {1} ( Cu) $ is fulfilled. For every Orlicz space $ L _ {M} ^ \star $ the inclusions $ L _ \infty \subset L _ {M} ^ \star \subset L _ {1} $ hold. Every summable function belongs to some Orlicz space.

The space $ L _ {M} ^ \star $ is separable if and only if $ M( u) $ satisfies the $ \Delta _ {2} $- condition (cf. Orlicz class). In general, $ L _ \infty $ is not dense in $ L _ {M} ^ \star $, and the closure of $ L _ \infty $ in $ L _ {M} ^ \star $ is denoted by $ E _ {M} $ and is always separable. If $ x \in L _ {M} ^ \star $, then

$$ \lim\limits _ {\tau \rightarrow \infty } \sup _ {mes( E) = \tau } \| x {\chi _ {E} } \| _ {M} = \rho ( x, E _ {M} ), $$

where

$$ \chi _ {E} ( t) = \left \{ \begin{array}{ll} 1 , & t \in E, \\ 0, & t \notin E. \\ \end{array} \right .$$

If $ M( u) $ and $ N( u) $ are complementary $ N $- functions and $ x \in L _ {M} ^ \star $, $ y \in L _ {M} ^ \star $, then the following analogue of the Hölder inequality holds:

$$ \int\limits _ { G } x( t) y( t) dt \leq \| x \| _ {(} M) \| y \| _ {(} N) , $$

where $ \| x \| _ {(} M) $ is the Luxemburg norm. Every continuous linear functional $ f $ on $ E _ {M} $ can be represented in the form

$$ f( x) = \int\limits _ { G } x( t) y( t) dt, $$

where $ y \in L _ {N} $ and $ \| f \| = \| y \| _ {(} N) $.

The compactness criteria of M. Riesz and A.N. Kolmogorov for the spaces $ L _ {p} $ can also be applied to $ E _ {M} $. The following conditions are equivalent:

1) the space $ L _ {M} ^ \star $ is reflexive;

2) $ M( u) $ and $ N( u) $ satisfy the $ \Delta _ {2} $- condition;

3) an unconditional basis exists in $ L _ {M} ^ \star $;

4) the Haar system forms an unconditional basis in $ L _ {M} ^ \star $;

5) the trigonometric system is a basis in $ L _ {M} ^ \star $ and the Haar system is a basis in $ E _ {M} $.

A sequence space $ l _ {M} ^ \star $ is defined in the same way, but the properties of $ l _ {M} ^ \star $ depend on the asymptotics of the function $ M( u) $ at 0. Many of the geometric properties of $ L _ {M} ^ \star $ and $ l _ {M} ^ \star $ are studied in [5]; for example, for any function $ M( u) $, the set of all $ p $ such that $ l _ {p} $ is isomorphically imbeddable in $ L _ {M} ^ \star $ can be found.

Orlicz spaces are used in the study of the properties of integral operators, the theory of differentiable functions of several variables and in other areas of analysis.

References

[1] W. Orlicz, "Ueber eine gewisse Klasse von Räumen vom Typus " Bull. Intern. Acad. Pol. Ser. A , 8/9 (1932) pp. 207–220
[2] M.A. Krasnosel'skii, Ya.B. Rutitskii, "Convex functions and Orlicz spaces" , Noordhoff (1961) (Translated from Russian)
[3] V.F. Gaposhkin, "Existence of absolute bases in Orlicz spaces" Funct. Anal. Appl. , 1 : 4 (1967) pp. 278–284 Funkts. Anal. i Prilozhen. , 1 : 4 (1967) pp. 26–32
[4] S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)
[5] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)

Comments

References

[a1] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
How to Cite This Entry:
Orlicz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orlicz_space&oldid=49344
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article