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''vector lattice''
 
''vector lattice''
  
A real partially ordered vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822901.png" /> (cf. [[Partially ordered set|Partially ordered set]]; [[Vector space|Vector space]]) in which
+
A real partially ordered vector space $  X $(
 +
cf. [[Partially ordered set|Partially ordered set]]; [[Vector space|Vector space]]) in which
  
1) the vector space structure and the partial order are compatible. i.e. from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822903.png" /> follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822904.png" /> and from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822908.png" /> follows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r0822909.png" />;
+
1) the vector space structure and the partial order are compatible. i.e. from $  x, y, z \in X $
 +
and $  x < y $
 +
follows that $  x+ z < y+ z $
 +
and from $  x \in X $,
 +
$  x > 0 $,  
 +
$  \lambda \in \mathbf R $,  
 +
$  \lambda > 0 $
 +
follows  $  \lambda x > 0 $;
  
2) for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229010.png" /> there exists <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229011.png" />. In particular, the supremum and infimum of any finite set exist.
+
2) for any two elements $  x, y $
 +
there exists $  \sup ( x, y) \in X $.  
 +
In particular, the supremum and infimum of any finite set exist.
  
In Soviet scientific literature Riesz spaces are usually called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229013.png" />-lineals. Such spaces were first introduced by F. Riesz in 1928.
+
In Soviet scientific literature Riesz spaces are usually called $  K $-
 +
lineals. Such spaces were first introduced by F. Riesz in 1928.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229014.png" /> of real continuous functions with the pointwise order is an example of a Riesz space. For any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229015.png" /> of a Riesz space one can define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229018.png" />. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229019.png" />. In Riesz spaces one can introduce two types of convergence of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229020.png" />. Order convergence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229022.png" />-convergence: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229023.png" /> if there exist a monotone increasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229024.png" /> and a monotone decreasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229027.png" />. Relative uniform convergence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229029.png" />-convergence: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229030.png" /> if there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229031.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229032.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229034.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229036.png" />-convergence is also called convergence with a regulator). The concepts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229037.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229038.png" />-convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229039.png" />.
+
The space $  C[ a, b] $
 +
of real continuous functions with the pointwise order is an example of a Riesz space. For any element $  x $
 +
of a Riesz space one can define $  x _ {+} = \sup ( x, 0) $,
 +
$  x _ {-} = \sup (- x, 0) $
 +
and $  | x | = x _ {+} + x _ {-} $.  
 +
It turns out that $  x = x _ {+} - x _ {-} $.  
 +
In Riesz spaces one can introduce two types of convergence of a sequence $  \{ x _ {n} \} $.  
 +
Order convergence, $  o $-
 +
convergence: $  x _ {n} \rightarrow  ^ {o} x _ {0} $
 +
if there exist a monotone increasing sequence $  \{ y _ {n} \} $
 +
and a monotone decreasing sequence $  \{ z _ {n} \} $
 +
such that $  y _ {n} \leq  x _ {n} \leq  z _ {n} $
 +
and $  \sup  y _ {n} = \inf  z _ {n} = x _ {0} $.  
 +
Relative uniform convergence, r $-
 +
convergence: $  x _ {n} \rightarrow  ^ {r} x _ {0} $
 +
if there exists an element $  u > 0 $
 +
such that for any $  \epsilon > 0 $
 +
there exists an $  n _ {0} $
 +
such that $  | x _ {n} - x _ {0} | < \epsilon u $
 +
for $  n \geq  n _ {0} $(
 +
r $-
 +
convergence is also called convergence with a regulator). The concepts of $  o $-  
 +
and r $-
 +
convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets $  \{ x _  \alpha  \} _ {\alpha \in \mathfrak A }  \subset  X $.
  
A Riesz space is called Archimedean if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229042.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229043.png" />. In Archimedean Riesz spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229045.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229046.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229048.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229049.png" />-convergence implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229050.png" />-convergence.
+
A Riesz space is called Archimedean if $  x, y \in X $
 +
and $  nx \leq  y $
 +
for $  n = 1, 2 , . . . $
 +
imply $  x \leq  0 $.  
 +
In Archimedean Riesz spaces, $  \lambda _ {n} \rightarrow \lambda _ {0} $
 +
and $  x _ {n} \rightarrow  ^ {o} x _ {0} $
 +
imply $  \lambda _ {n} x _ {n} \rightarrow  ^ {o} \lambda _ {0} x _ {0} $(
 +
$  \lambda _ {n} , \lambda _ {0} \in \mathbf R $,
 +
$  x _ {n} , x _ {0} \in X $),  
 +
and r $-
 +
convergence implies $  o $-
 +
convergence.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  "Sur la décomposition des opérations fonctionelles linéaires" , ''Atti congress. internaz. mathematici (Bologna, 1928)'' , '''3''' , Zanichelli  (1930)  pp. 143–148</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.Z. Vulikh,  "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  "Sur la décomposition des opérations fonctionelles linéaires" , ''Atti congress. internaz. mathematici (Bologna, 1928)'' , '''3''' , Zanichelli  (1930)  pp. 143–148</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.Z. Vulikh,  "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A Riesz subspace of a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229051.png" /> is a linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229055.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229056.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229057.png" /> (where the sup and inf are those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229058.png" />). A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229060.png" /> that is an order ideal, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229063.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229064.png" />, is called a Riesz ideal. Such subspaces are called sublineals and normal sublineals in the Soviet literature. A band is a Riesz ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229065.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229066.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229067.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229068.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229069.png" /> exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229070.png" />. A band is often called a component in the Soviet literature.
+
A Riesz subspace of a Riesz space $  L $
 +
is a linear subspace $  K $
 +
of $  L $
 +
such that $  \sup ( f, g) = f \lor g $
 +
and $  \inf ( f, g) = f \wedge g $
 +
are in $  K $
 +
whenever $  f, g \in K $(
 +
where the sup and inf are those of $  L $).  
 +
A subspace $  A $
 +
of $  L $
 +
that is an order ideal, i.e. $  f \in A $,  
 +
$  g \in L $,  
 +
$  | g | \leq  | f | $
 +
imply that $  g \in A $,  
 +
is called a Riesz ideal. Such subspaces are called sublineals and normal sublineals in the Soviet literature. A band is a Riesz ideal $  A $
 +
such that $  \sup  D $
 +
in $  A $
 +
for $  D \subset  A $
 +
if $  \sup  D $
 +
exists in $  L $.  
 +
A band is often called a component in the Soviet literature.
  
A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229071.png" /> from a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229072.png" /> to a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229073.png" /> is called positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229074.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229076.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229077.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229078.png" /> is called order bounded if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229079.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229081.png" />. The linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229082.png" /> is called order bounded if it takes order-bounded sets to order-bounded sets. Taking the positive operators as the positive cone defines an order structure on the space of order-bounded operators, turning it into a Dedekind-complete Riesz space (the Freudenthal–Kantorovich theorem). Recall that a lattice is Dedekind complete if every subset bounded from below (respectively above) has an inf (respectively sup). A positive operator is order bounded, and so are differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229083.png" /> of positive operators, which are called regular operators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229084.png" /> is Dedekind complete, the converse holds: Every order-bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229085.png" /> admits a Jordan decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229086.png" /> as a difference of two positive operators.
+
A linear operator $  T $
 +
from a Riesz space $  L $
 +
to a Riesz space $  M $
 +
is called positive if $  Tf \geq  0 $
 +
for all $  f \geq  0 $,  
 +
$  f \in L $.  
 +
A set $  D $
 +
in $  L $
 +
is called order bounded if there exist $  f, g \in L $
 +
such that $  f \leq  d \leq  g $
 +
for all $  d \in D $.  
 +
The linear operator $  T $
 +
is called order bounded if it takes order-bounded sets to order-bounded sets. Taking the positive operators as the positive cone defines an order structure on the space of order-bounded operators, turning it into a Dedekind-complete Riesz space (the Freudenthal–Kantorovich theorem). Recall that a lattice is Dedekind complete if every subset bounded from below (respectively above) has an inf (respectively sup). A positive operator is order bounded, and so are differences $  T _ {1} - T _ {2} $
 +
of positive operators, which are called regular operators. If $  M $
 +
is Dedekind complete, the converse holds: Every order-bounded operator $  T $
 +
admits a Jordan decomposition $  T = T _ {1} - T _ {2} $
 +
as a difference of two positive operators.
  
A norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229087.png" /> on a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229088.png" /> is a Riesz norm if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229089.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229090.png" />. A Riesz semi-norm is a semi-norm with the same compatibility conditions. A Riesz space with a Riesz norm is a normed Riesz space. A norm-complete normed Riesz space is a [[Banach lattice|Banach lattice]]. An order-bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229091.png" /> from a Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229092.png" /> to a Dedekind-complete normed Riesz space is norm bounded.
+
A norm $  \| \cdot \| $
 +
on a Riesz space $  L $
 +
is a Riesz norm if $  | f | \leq  | g | $
 +
implies $  \| f \| \leq  \| g \| $.  
 +
A Riesz semi-norm is a semi-norm with the same compatibility conditions. A Riesz space with a Riesz norm is a normed Riesz space. A norm-complete normed Riesz space is a [[Banach lattice|Banach lattice]]. An order-bounded operator $  T $
 +
from a Banach lattice $  L $
 +
to a Dedekind-complete normed Riesz space is norm bounded.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229093.png" /> be the space of order-bounded operators from a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229094.png" /> to a Dedekind-complete Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229095.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229096.png" /> is called sequentially order continuous, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229098.png" />-order continuous, if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r08229099.png" /> (i.e. that is monotonically decreasing to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290100.png" />) it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290101.png" />; it is called order continuous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290102.png" /> for every downwards directed system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290104.png" /> (cf. [[Directed set|Directed set]]). The Soviet terminology for order-continuous and sequentially order-continuous linear operators is o-linear and (o)-linear. The order-continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290105.png" />-order continuous operators are bands in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290106.png" />. The order dual of a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290107.png" /> is the space of order-bounded operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290108.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290109.png" />. The result that this order dual is Dedekind complete goes back to F. Riesz.
+
Let $  T _ {b} ( L, M) $
 +
be the space of order-bounded operators from a Riesz space $  L $
 +
to a Dedekind-complete Riesz space $  M $.  
 +
$  T \in T _ {b} ( L, M) $
 +
is called sequentially order continuous, or $  \tau $-
 +
order continuous, if for every sequence $  u _ {n} \downarrow 0 $(
 +
i.e. that is monotonically decreasing to 0 $)  
 +
it follows that $  \inf  | Tu _ {n} | = 0 $;  
 +
it is called order continuous if $  \inf  | Tu _  \tau  | = 0 $
 +
for every downwards directed system $  u _  \tau  \rightarrow 0 $
 +
in $  L $(
 +
cf. [[Directed set|Directed set]]). The Soviet terminology for order-continuous and sequentially order-continuous linear operators is o-linear and (o)-linear. The order-continuous and $  \tau $-
 +
order continuous operators are bands in $  T _ {b} ( L, M) $.  
 +
The order dual of a Riesz space $  L $
 +
is the space of order-bounded operators of $  L $
 +
into $  \mathbf R $.  
 +
The result that this order dual is Dedekind complete goes back to F. Riesz.
  
 
There is a second important concept of duality in Riesz space theory, reminiscent of both linear duality and the algebraic geometric duality:  "ideals  zero sets" , that is basic to scheme theory. It is called Baker–Benyon duality (see the volume with supplementary articles).
 
There is a second important concept of duality in Riesz space theory, reminiscent of both linear duality and the algebraic geometric duality:  "ideals  zero sets" , that is basic to scheme theory. It is called Baker–Benyon duality (see the volume with supplementary articles).
  
In the theory of linear topological spaces (cf. [[Topological vector space|Topological vector space]]) the following criterion for boundedness of a set is used: A set B is bounded (in this theory) if and only if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290111.png" />, and sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290112.png" /> converging to zero, one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290113.png" /> converges to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290114.png" />. The question arises whether order-bounded sets in a Riesz space can be characterized in this way, using instead order convergence of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290115.png" /> to zero. For arbitrary Dedekind-complete Riesz spaces this need not be true. The Dedekind-complete Riesz spaces for which this criterion holds are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290117.png" />-spaces.
+
In the theory of linear topological spaces (cf. [[Topological vector space|Topological vector space]]) the following criterion for boundedness of a set is used: A set B is bounded (in this theory) if and only if for every sequence $  ( x _ {n} ) _ {n} $,  
 +
$  x _ {n} \in B $,  
 +
and sequence of real numbers $  ( \lambda _ {n} ) _ {n} $
 +
converging to zero, one has that $  ( \lambda _ {n} x _ {n} ) _ {n} $
 +
converges to zero as $  n \rightarrow \infty $.  
 +
The question arises whether order-bounded sets in a Riesz space can be characterized in this way, using instead order convergence of the $  ( \lambda _ {n} x _ {n} ) _ {n} $
 +
to zero. For arbitrary Dedekind-complete Riesz spaces this need not be true. The Dedekind-complete Riesz spaces for which this criterion holds are called $  K  ^ {+} $-
 +
spaces.
  
Let now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290118.png" /> be a normed space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290119.png" /> a Dedekind-complete Riesz space. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290120.png" /> is called bo-linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290121.png" /> in norm implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290122.png" /> in order convergence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290123.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290124.png" />-space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290125.png" /> is bo-linear if and only if the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290126.png" /> of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290127.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290128.png" /> is order bounded. The element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290129.png" /> defined by
+
Let now $  L $
 +
be a normed space and $  M $
 +
a Dedekind-complete Riesz space. A linear operator $  U : L \rightarrow M $
 +
is called bo-linear if $  x _ {n} \rightarrow x $
 +
in norm implies $  U x _ {n} \rightarrow U x $
 +
in order convergence. If $  M $
 +
is a $  K  ^ {+} $-
 +
space, then $  U : L \rightarrow M $
 +
is bo-linear if and only if the image $  U( S) $
 +
of the unit sphere $  S $
 +
in $  L $
 +
is order bounded. The element of $  M $
 +
defined by
  
<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/r/r082/r082290/r082290130.png" /></td> </tr></table>
+
$$
 +
| U |  = \sup _ {\| x \| \leq  1 }  | U x |
 +
$$
  
is then called an abstract norm for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290131.png" />.
+
is then called an abstract norm for the operator $  U $.
  
There is a variety of Riesz space analogues of Hahn–Banach type extension and existence theorems. A selection follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290132.png" /> be a normed space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290133.png" /> a linear subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290135.png" /> a bo-linear operator into a Dedekind-complete Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290136.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290137.png" /> possesses an abstract norm. Then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290138.png" /> admits a bo-linear extension to all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290139.png" /> with the same abstract norm. This is one of the Kantorovich extension theorems. Another extension theorem for Riesz spaces, also due to B.Z. Kantorovich, concerns the extension of positive operators: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290140.png" /> be a Riesz space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290141.png" /> a linear subset that majorizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290142.png" />, i.e. for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290143.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290144.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290145.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290146.png" /> be a positive additive operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290147.png" /> into a Dedekind-complete Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290148.png" />. Then there exists an additive and positive extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290149.png" /> to all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290150.png" />. Using these and/or related extension theorems one can show that a positive linear functional on a Riesz subspace of a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290151.png" /> that is majorized by a Riesz semi-norm can be extended to a positive functional on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290152.png" />, a result which in turn serves to discuss when the order dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290153.png" /> is at least non-zero.
+
There is a variety of Riesz space analogues of Hahn–Banach type extension and existence theorems. A selection follows. Let $  L $
 +
be a normed space, $  E $
 +
a linear subset of $  L $
 +
and $  U: E \rightarrow M $
 +
a bo-linear operator into a Dedekind-complete Riesz space $  M $.  
 +
Suppose that $  U $
 +
possesses an abstract norm. Then the operator $  U $
 +
admits a bo-linear extension to all of $  L $
 +
with the same abstract norm. This is one of the Kantorovich extension theorems. Another extension theorem for Riesz spaces, also due to B.Z. Kantorovich, concerns the extension of positive operators: Let $  X $
 +
be a Riesz space and $  E $
 +
a linear subset that majorizes $  X $,  
 +
i.e. for every $  x \in X $
 +
there is an $  e \in E $
 +
such that $  | x | \leq  e $.  
 +
Let $  U : E \rightarrow M $
 +
be a positive additive operator from $  E $
 +
into a Dedekind-complete Riesz space $  Y $.  
 +
Then there exists an additive and positive extension of $  U $
 +
to all of $  X $.  
 +
Using these and/or related extension theorems one can show that a positive linear functional on a Riesz subspace of a Riesz space $  L $
 +
that is majorized by a Riesz semi-norm can be extended to a positive functional on all of $  L $,  
 +
a result which in turn serves to discuss when the order dual of $  L $
 +
is at least non-zero.
  
 
Examples of Riesz spaces are provided by spaces of real-valued functions on a topological space (possibly, extended real functions), where the order is defined pointwise. As in the case of, e.g., Banach algebras (cf. [[Banach algebra|Banach algebra]]), where the Gel'fand representation provides an answer, one asks whether an arbitrary Riesz space can be seen as a space of real-valued functions on a suitable space (of ideals). The answer for Riesz spaces is given by the [[Yosida representation theorem|Yosida representation theorem]] and its relatives.
 
Examples of Riesz spaces are provided by spaces of real-valued functions on a topological space (possibly, extended real functions), where the order is defined pointwise. As in the case of, e.g., Banach algebras (cf. [[Banach algebra|Banach algebra]]), where the Gel'fand representation provides an answer, one asks whether an arbitrary Riesz space can be seen as a space of real-valued functions on a suitable space (of ideals). The answer for Riesz spaces is given by the [[Yosida representation theorem|Yosida representation theorem]] and its relatives.
  
In the integration theory (of real functions) a basic role is played by such operations as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290155.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290157.png" />, which makes at least potentially credible that Riesz spaces might provide a suitable abstract setting for integration theory. This is indeed the case in the form of the Freudenthal spectral theorem, which will be discussed below.
+
In the integration theory (of real functions) a basic role is played by such operations as $  f = f ^ { + } - f ^ { - } $,  
 +
$  | f | = f ^ { + } + f ^ { - } $,  
 +
where $  f ^ { + } ( x) = \max ( f ( x) , 0) $,
 +
$  f ^ { - } = \max (- f( x), 0) $,  
 +
which makes at least potentially credible that Riesz spaces might provide a suitable abstract setting for integration theory. This is indeed the case in the form of the Freudenthal spectral theorem, which will be discussed below.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290158.png" /> be a lattice with zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290159.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290160.png" /> be a non-empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290161.png" />; the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290162.png" /> that are disjoint from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290163.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290164.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290165.png" />, is called the disjoint complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290166.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290167.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290168.png" />. In a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290169.png" /> two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290170.png" /> are called disjoint if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290171.png" />. (This agrees with the previous definition if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290173.png" /> are both positive.)
+
Let $  X $
 +
be a lattice with zero, 0 $.  
 +
Let $  Y $
 +
be a non-empty subset of $  X $;  
 +
the set of $  x \in X $
 +
that are disjoint from $  Y $,  
 +
i.e. $  x \wedge y = 0 $
 +
for all $  y \in Y $,  
 +
is called the disjoint complement of $  Y $
 +
in $  X $
 +
and is denoted by $  Y  ^ {d} $.  
 +
In a Riesz space $  L $
 +
two elements $  f , g $
 +
are called disjoint if $  | f | \wedge | g | = 0 $.  
 +
(This agrees with the previous definition if $  f $
 +
and $  g $
 +
are both positive.)
  
Given a band <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290174.png" /> in a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290175.png" />, the disjoint complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290176.png" /> is also a band. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290177.png" /> is Dedekind complete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290178.png" />. In general, a band <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290179.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290180.png" /> is called a projection band. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band. Thus, a Dedekind-complete Riesz space has the projection property and, a fortiori, the principal projection property.
+
Given a band $  A $
 +
in a Riesz space $  L $,  
 +
the disjoint complement $  A  ^ {d} $
 +
is also a band. If $  L $
 +
is Dedekind complete, $  L = A \oplus A  ^ {d} $.  
 +
In general, a band $  A $
 +
such that $  L= A \oplus A  ^ {d} $
 +
is called a projection band. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band. Thus, a Dedekind-complete Riesz space has the projection property and, a fortiori, the principal projection property.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290181.png" /> be a Riesz space with the principal projection property, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290182.png" /> be a non-zero positive element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290183.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290184.png" /> be an element in the band generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290185.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290186.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290187.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290188.png" /> be the component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290189.png" /> in the band <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290190.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290191.png" /> under the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290192.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290193.png" /> is called the spectral system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290194.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290195.png" />. Now suppose there is a finite interval such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290196.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290197.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290198.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290199.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290200.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290201.png" />. For every partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290202.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290204.png" /> one forms the lower and upper sums
+
Let $  L $
 +
be a Riesz space with the principal projection property, let $  e $
 +
be a non-zero positive element of $  L $
 +
and let $  f $
 +
be an element in the band generated by $  e $.  
 +
Let $  u _  \alpha  = \sup ( \alpha e - f, 0 ) $
 +
for $  - \infty < \alpha < \infty $,  
 +
and let $  p _  \alpha  $
 +
be the component of $  e $
 +
in the band $  B _  \alpha  $
 +
generated by $  u _  \alpha  $
 +
under the decomposition $  L = B _  \alpha  \oplus B _  \alpha  ^ {d} $.  
 +
The set $  ( p _  \alpha  ) _  \alpha  $
 +
is called the spectral system of $  f $
 +
with respect to $  e $.  
 +
Now suppose there is a finite interval such that $  ae \leq  f \leq  ( b - \epsilon ) e $
 +
for some $  \epsilon > 0 $.  
 +
Then $  p _  \alpha  = 0 $
 +
for $  \alpha \leq  a $
 +
and $  p _  \alpha  = e $
 +
for $  \alpha \geq  b $.  
 +
For every partition $  \pi $:
 +
$  a = \alpha _ {0} < \alpha _ {1} < \dots < \alpha _ {n} = b $
 +
of $  [ a , b] $
 +
one forms the lower and upper sums
  
<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/r/r082/r082290/r082290205.png" /></td> </tr></table>
+
$$
 +
s ( \pi , f  )  = \
 +
\sum _ { k= } 1 ^ { n }  \alpha _ {k-} 1
 +
( p _ {\alpha _ {k}  } - p _ {\alpha _ {k-} 1 }  ),
 +
$$
  
<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/r/r082/r082290/r082290206.png" /></td> </tr></table>
+
$$
 +
u ( \pi , f  )  = \sum _ { k= } 1 ^ { n }  \alpha _ {k} ( p _ {\alpha _ {k}  } - p _ {\alpha _ {k-} 1 }  ) .
 +
$$
  
One then has the following result in abstract integration theory, known as the Freudenthal spectral theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290210.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290212.png" /> be as above. Then
+
One then has the following result in abstract integration theory, known as the Freudenthal spectral theorem. Let $  L $,  
 +
$  e $,  
 +
$  f $,  
 +
$  a $,  
 +
$  b $,  
 +
$  \epsilon $
 +
be as above. Then
  
<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/r/r082/r082290/r082290213.png" /></td> </tr></table>
+
$$
 +
\sup _  \pi  s ( \pi , f  )  = \
 +
= \inf _  \pi  u( \pi , f  ).
 +
$$
  
In the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290214.png" /> is a Riesz space of real-valued functions on a space (especially a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290215.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290216.png" />, this spectral theorem expresses approximation properties of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082290/r082290217.png" /> by  "step functions" . The [[Radon–Nikodým theorem|Radon–Nikodým theorem]] in measure theory and the [[Poisson formula|Poisson formula]] for bounded harmonic functions on an open disc are special cases of the spectral theorem. The Freudenthal spectral theorem was one of the starting points of Riesz space theory.
+
In the case that $  L $
 +
is a Riesz space of real-valued functions on a space (especially a subset of $  \mathbf R $)  
 +
and $  e( x) = 1 $,  
 +
this spectral theorem expresses approximation properties of functions in $  L $
 +
by  "step functions" . The [[Radon–Nikodým theorem|Radon–Nikodým theorem]] in measure theory and the [[Poisson formula|Poisson formula]] for bounded harmonic functions on an open disc are special cases of the spectral theorem. The Freudenthal spectral theorem was one of the starting points of Riesz space theory.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.H. Schaefer,  "Banach lattices and positive operators" , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. de Jonge,  A.C.M. van Rooy,  "Introduction to Riesz spaces" , ''Tracts'' , '''8''' , Math. Centre  (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.Z. Kantorovich,  B.Z. Vulikh,  A.G. Pinsker,  "Functional analysis in partially ordered spaces" , Moscow  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Freudenthal,  "Teilweise geordneten Moduln"  ''Proc. Royal Acad. Sci. Amsterdam'' , '''39'''  (1936)  pp. 641–651</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Nakano,  "Modern spectral theory" , Maruzen  (1950)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.H. Schaefer,  "Banach lattices and positive operators" , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. de Jonge,  A.C.M. van Rooy,  "Introduction to Riesz spaces" , ''Tracts'' , '''8''' , Math. Centre  (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.Z. Kantorovich,  B.Z. Vulikh,  A.G. Pinsker,  "Functional analysis in partially ordered spaces" , Moscow  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Freudenthal,  "Teilweise geordneten Moduln"  ''Proc. Royal Acad. Sci. Amsterdam'' , '''39'''  (1936)  pp. 641–651</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Nakano,  "Modern spectral theory" , Maruzen  (1950)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


vector lattice

A real partially ordered vector space $ X $( cf. Partially ordered set; Vector space) in which

1) the vector space structure and the partial order are compatible. i.e. from $ x, y, z \in X $ and $ x < y $ follows that $ x+ z < y+ z $ and from $ x \in X $, $ x > 0 $, $ \lambda \in \mathbf R $, $ \lambda > 0 $ follows $ \lambda x > 0 $;

2) for any two elements $ x, y $ there exists $ \sup ( x, y) \in X $. In particular, the supremum and infimum of any finite set exist.

In Soviet scientific literature Riesz spaces are usually called $ K $- lineals. Such spaces were first introduced by F. Riesz in 1928.

The space $ C[ a, b] $ of real continuous functions with the pointwise order is an example of a Riesz space. For any element $ x $ of a Riesz space one can define $ x _ {+} = \sup ( x, 0) $, $ x _ {-} = \sup (- x, 0) $ and $ | x | = x _ {+} + x _ {-} $. It turns out that $ x = x _ {+} - x _ {-} $. In Riesz spaces one can introduce two types of convergence of a sequence $ \{ x _ {n} \} $. Order convergence, $ o $- convergence: $ x _ {n} \rightarrow ^ {o} x _ {0} $ if there exist a monotone increasing sequence $ \{ y _ {n} \} $ and a monotone decreasing sequence $ \{ z _ {n} \} $ such that $ y _ {n} \leq x _ {n} \leq z _ {n} $ and $ \sup y _ {n} = \inf z _ {n} = x _ {0} $. Relative uniform convergence, $ r $- convergence: $ x _ {n} \rightarrow ^ {r} x _ {0} $ if there exists an element $ u > 0 $ such that for any $ \epsilon > 0 $ there exists an $ n _ {0} $ such that $ | x _ {n} - x _ {0} | < \epsilon u $ for $ n \geq n _ {0} $( $ r $- convergence is also called convergence with a regulator). The concepts of $ o $- and $ r $- convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets $ \{ x _ \alpha \} _ {\alpha \in \mathfrak A } \subset X $.

A Riesz space is called Archimedean if $ x, y \in X $ and $ nx \leq y $ for $ n = 1, 2 , . . . $ imply $ x \leq 0 $. In Archimedean Riesz spaces, $ \lambda _ {n} \rightarrow \lambda _ {0} $ and $ x _ {n} \rightarrow ^ {o} x _ {0} $ imply $ \lambda _ {n} x _ {n} \rightarrow ^ {o} \lambda _ {0} x _ {0} $( $ \lambda _ {n} , \lambda _ {0} \in \mathbf R $, $ x _ {n} , x _ {0} \in X $), and $ r $- convergence implies $ o $- convergence.

References

[1] F. Riesz, "Sur la décomposition des opérations fonctionelles linéaires" , Atti congress. internaz. mathematici (Bologna, 1928) , 3 , Zanichelli (1930) pp. 143–148
[2] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
[3] B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)

Comments

A Riesz subspace of a Riesz space $ L $ is a linear subspace $ K $ of $ L $ such that $ \sup ( f, g) = f \lor g $ and $ \inf ( f, g) = f \wedge g $ are in $ K $ whenever $ f, g \in K $( where the sup and inf are those of $ L $). A subspace $ A $ of $ L $ that is an order ideal, i.e. $ f \in A $, $ g \in L $, $ | g | \leq | f | $ imply that $ g \in A $, is called a Riesz ideal. Such subspaces are called sublineals and normal sublineals in the Soviet literature. A band is a Riesz ideal $ A $ such that $ \sup D $ in $ A $ for $ D \subset A $ if $ \sup D $ exists in $ L $. A band is often called a component in the Soviet literature.

A linear operator $ T $ from a Riesz space $ L $ to a Riesz space $ M $ is called positive if $ Tf \geq 0 $ for all $ f \geq 0 $, $ f \in L $. A set $ D $ in $ L $ is called order bounded if there exist $ f, g \in L $ such that $ f \leq d \leq g $ for all $ d \in D $. The linear operator $ T $ is called order bounded if it takes order-bounded sets to order-bounded sets. Taking the positive operators as the positive cone defines an order structure on the space of order-bounded operators, turning it into a Dedekind-complete Riesz space (the Freudenthal–Kantorovich theorem). Recall that a lattice is Dedekind complete if every subset bounded from below (respectively above) has an inf (respectively sup). A positive operator is order bounded, and so are differences $ T _ {1} - T _ {2} $ of positive operators, which are called regular operators. If $ M $ is Dedekind complete, the converse holds: Every order-bounded operator $ T $ admits a Jordan decomposition $ T = T _ {1} - T _ {2} $ as a difference of two positive operators.

A norm $ \| \cdot \| $ on a Riesz space $ L $ is a Riesz norm if $ | f | \leq | g | $ implies $ \| f \| \leq \| g \| $. A Riesz semi-norm is a semi-norm with the same compatibility conditions. A Riesz space with a Riesz norm is a normed Riesz space. A norm-complete normed Riesz space is a Banach lattice. An order-bounded operator $ T $ from a Banach lattice $ L $ to a Dedekind-complete normed Riesz space is norm bounded.

Let $ T _ {b} ( L, M) $ be the space of order-bounded operators from a Riesz space $ L $ to a Dedekind-complete Riesz space $ M $. $ T \in T _ {b} ( L, M) $ is called sequentially order continuous, or $ \tau $- order continuous, if for every sequence $ u _ {n} \downarrow 0 $( i.e. that is monotonically decreasing to $ 0 $) it follows that $ \inf | Tu _ {n} | = 0 $; it is called order continuous if $ \inf | Tu _ \tau | = 0 $ for every downwards directed system $ u _ \tau \rightarrow 0 $ in $ L $( cf. Directed set). The Soviet terminology for order-continuous and sequentially order-continuous linear operators is o-linear and (o)-linear. The order-continuous and $ \tau $- order continuous operators are bands in $ T _ {b} ( L, M) $. The order dual of a Riesz space $ L $ is the space of order-bounded operators of $ L $ into $ \mathbf R $. The result that this order dual is Dedekind complete goes back to F. Riesz.

There is a second important concept of duality in Riesz space theory, reminiscent of both linear duality and the algebraic geometric duality: "ideals zero sets" , that is basic to scheme theory. It is called Baker–Benyon duality (see the volume with supplementary articles).

In the theory of linear topological spaces (cf. Topological vector space) the following criterion for boundedness of a set is used: A set B is bounded (in this theory) if and only if for every sequence $ ( x _ {n} ) _ {n} $, $ x _ {n} \in B $, and sequence of real numbers $ ( \lambda _ {n} ) _ {n} $ converging to zero, one has that $ ( \lambda _ {n} x _ {n} ) _ {n} $ converges to zero as $ n \rightarrow \infty $. The question arises whether order-bounded sets in a Riesz space can be characterized in this way, using instead order convergence of the $ ( \lambda _ {n} x _ {n} ) _ {n} $ to zero. For arbitrary Dedekind-complete Riesz spaces this need not be true. The Dedekind-complete Riesz spaces for which this criterion holds are called $ K ^ {+} $- spaces.

Let now $ L $ be a normed space and $ M $ a Dedekind-complete Riesz space. A linear operator $ U : L \rightarrow M $ is called bo-linear if $ x _ {n} \rightarrow x $ in norm implies $ U x _ {n} \rightarrow U x $ in order convergence. If $ M $ is a $ K ^ {+} $- space, then $ U : L \rightarrow M $ is bo-linear if and only if the image $ U( S) $ of the unit sphere $ S $ in $ L $ is order bounded. The element of $ M $ defined by

$$ | U | = \sup _ {\| x \| \leq 1 } | U x | $$

is then called an abstract norm for the operator $ U $.

There is a variety of Riesz space analogues of Hahn–Banach type extension and existence theorems. A selection follows. Let $ L $ be a normed space, $ E $ a linear subset of $ L $ and $ U: E \rightarrow M $ a bo-linear operator into a Dedekind-complete Riesz space $ M $. Suppose that $ U $ possesses an abstract norm. Then the operator $ U $ admits a bo-linear extension to all of $ L $ with the same abstract norm. This is one of the Kantorovich extension theorems. Another extension theorem for Riesz spaces, also due to B.Z. Kantorovich, concerns the extension of positive operators: Let $ X $ be a Riesz space and $ E $ a linear subset that majorizes $ X $, i.e. for every $ x \in X $ there is an $ e \in E $ such that $ | x | \leq e $. Let $ U : E \rightarrow M $ be a positive additive operator from $ E $ into a Dedekind-complete Riesz space $ Y $. Then there exists an additive and positive extension of $ U $ to all of $ X $. Using these and/or related extension theorems one can show that a positive linear functional on a Riesz subspace of a Riesz space $ L $ that is majorized by a Riesz semi-norm can be extended to a positive functional on all of $ L $, a result which in turn serves to discuss when the order dual of $ L $ is at least non-zero.

Examples of Riesz spaces are provided by spaces of real-valued functions on a topological space (possibly, extended real functions), where the order is defined pointwise. As in the case of, e.g., Banach algebras (cf. Banach algebra), where the Gel'fand representation provides an answer, one asks whether an arbitrary Riesz space can be seen as a space of real-valued functions on a suitable space (of ideals). The answer for Riesz spaces is given by the Yosida representation theorem and its relatives.

In the integration theory (of real functions) a basic role is played by such operations as $ f = f ^ { + } - f ^ { - } $, $ | f | = f ^ { + } + f ^ { - } $, where $ f ^ { + } ( x) = \max ( f ( x) , 0) $, $ f ^ { - } = \max (- f( x), 0) $, which makes at least potentially credible that Riesz spaces might provide a suitable abstract setting for integration theory. This is indeed the case in the form of the Freudenthal spectral theorem, which will be discussed below.

Let $ X $ be a lattice with zero, $ 0 $. Let $ Y $ be a non-empty subset of $ X $; the set of $ x \in X $ that are disjoint from $ Y $, i.e. $ x \wedge y = 0 $ for all $ y \in Y $, is called the disjoint complement of $ Y $ in $ X $ and is denoted by $ Y ^ {d} $. In a Riesz space $ L $ two elements $ f , g $ are called disjoint if $ | f | \wedge | g | = 0 $. (This agrees with the previous definition if $ f $ and $ g $ are both positive.)

Given a band $ A $ in a Riesz space $ L $, the disjoint complement $ A ^ {d} $ is also a band. If $ L $ is Dedekind complete, $ L = A \oplus A ^ {d} $. In general, a band $ A $ such that $ L= A \oplus A ^ {d} $ is called a projection band. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band. Thus, a Dedekind-complete Riesz space has the projection property and, a fortiori, the principal projection property.

Let $ L $ be a Riesz space with the principal projection property, let $ e $ be a non-zero positive element of $ L $ and let $ f $ be an element in the band generated by $ e $. Let $ u _ \alpha = \sup ( \alpha e - f, 0 ) $ for $ - \infty < \alpha < \infty $, and let $ p _ \alpha $ be the component of $ e $ in the band $ B _ \alpha $ generated by $ u _ \alpha $ under the decomposition $ L = B _ \alpha \oplus B _ \alpha ^ {d} $. The set $ ( p _ \alpha ) _ \alpha $ is called the spectral system of $ f $ with respect to $ e $. Now suppose there is a finite interval such that $ ae \leq f \leq ( b - \epsilon ) e $ for some $ \epsilon > 0 $. Then $ p _ \alpha = 0 $ for $ \alpha \leq a $ and $ p _ \alpha = e $ for $ \alpha \geq b $. For every partition $ \pi $: $ a = \alpha _ {0} < \alpha _ {1} < \dots < \alpha _ {n} = b $ of $ [ a , b] $ one forms the lower and upper sums

$$ s ( \pi , f ) = \ \sum _ { k= } 1 ^ { n } \alpha _ {k-} 1 ( p _ {\alpha _ {k} } - p _ {\alpha _ {k-} 1 } ), $$

$$ u ( \pi , f ) = \sum _ { k= } 1 ^ { n } \alpha _ {k} ( p _ {\alpha _ {k} } - p _ {\alpha _ {k-} 1 } ) . $$

One then has the following result in abstract integration theory, known as the Freudenthal spectral theorem. Let $ L $, $ e $, $ f $, $ a $, $ b $, $ \epsilon $ be as above. Then

$$ \sup _ \pi s ( \pi , f ) = \ f = \inf _ \pi u( \pi , f ). $$

In the case that $ L $ is a Riesz space of real-valued functions on a space (especially a subset of $ \mathbf R $) and $ e( x) = 1 $, this spectral theorem expresses approximation properties of functions in $ L $ by "step functions" . The Radon–Nikodým theorem in measure theory and the Poisson formula for bounded harmonic functions on an open disc are special cases of the spectral theorem. The Freudenthal spectral theorem was one of the starting points of Riesz space theory.

References

[a1] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a2] H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
[a3] E. de Jonge, A.C.M. van Rooy, "Introduction to Riesz spaces" , Tracts , 8 , Math. Centre (1977)
[a4] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[a5] B.Z. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in partially ordered spaces" , Moscow (1950) (In Russian)
[a6] H. Freudenthal, "Teilweise geordneten Moduln" Proc. Royal Acad. Sci. Amsterdam , 39 (1936) pp. 641–651
[a7] H. Nakano, "Modern spectral theory" , Maruzen (1950)
How to Cite This Entry:
Riesz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_space&oldid=17299
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article