View source for Riesz space

''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

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" />;

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.

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.

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" />.

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.

====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>



====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 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 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.

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.

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.

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

<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>

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" />.

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.

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.

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.)

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.

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

<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>

<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>

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

<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>

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.

====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>