Riesz room

A Riesz space is a vector space with a lattice structure that is designed so that the linear and lattice structure are compatible. In 1928 this space was defined by Frigyes Riesz and therefore bears his name today.

definition

Let be a - vector space and a semi-ordered set. ${\displaystyle (X,+,\cdot )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (X,\leq )}$

Then a Riesz space is called if the following axioms are fulfilled: ${\displaystyle (X,+,\cdot ,\leq )}$

1. For all true: . ${\displaystyle f,g,h\in X}$ ${\displaystyle f\leq g\Rightarrow f+h\leq g+h}$
2. The following applies to all : and . ${\displaystyle f,g\in X}$ ${\displaystyle 0\leq a\in \mathbb {R} }$ ${\displaystyle f\leq g\Rightarrow a\cdot f\leq a\cdot g}$
3. ${\displaystyle (X,\leq )}$is an association .

Remarks

• 1. and 2. mean is an ordered vector space. ${\displaystyle (X,+,\cdot ,\leq )}$
• In the formulation of 2. it should be noted that both refer to and to , from the context it is usually clear which order relation is meant, so that additional indices are usually not used. ${\displaystyle \leq }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$
• 2. can also be replaced by the weaker requirement and . ${\displaystyle 0\leq a}$ ${\displaystyle \mathbf {0} \leq f\Rightarrow \mathbf {0} \leq a\cdot f}$
• If the association operations denote , it is convention that bind stronger than (rule in brackets). ${\displaystyle \land ,\lor }$ ${\displaystyle \land ,\lor }$ ${\displaystyle +,\cdot }$

First properties

For and the following calculation rules apply: ${\displaystyle f,g,h\in X}$ ${\displaystyle 0\leq a\in \mathbb {R} }$

• ${\displaystyle (f+h)\lor (g+h)=(f\lor g)+h}$ and ${\displaystyle (f+h)\land (g+h)=(f\land g)+h}$
• ${\displaystyle (af)\lor (ag)=a(f\lor g)}$ and ${\displaystyle (af)\land (ag)=a(f\land g)}$
• ${\displaystyle (-f)\lor (-g)=-(f\land g)}$ and ${\displaystyle (-f)\land (-g)=-(f\lor g)}$
• Be for . ${\displaystyle |f|:=f\lor (-f)}$ ${\displaystyle f\in X}$

Then and . ${\displaystyle f\lor g={\tfrac {1}{2}}(f+g+|f-g|)}$ ${\displaystyle f\land g={\tfrac {1}{2}}(f+g-|f-g|)}$

• ${\displaystyle (f\lor g)+(f\land g)=f+g}$ and ${\displaystyle (f\lor g)-(f\land g)=|f-g|}$
• ${\displaystyle (f\lor g)\land h=(f\land h)\lor (g\land h)}$ and ${\displaystyle (f\land g)\lor h=(f\lor h)\land (g\lor h)}$

This means that every Riesz area is a distributive association .

Examples

• The real numbers with the usual arrangement form a Riesz space. ${\displaystyle \mathbb {R} }$ ${\displaystyle \leq }$
• The one with a component arrangement forms a Riesz room. ${\displaystyle \mathbb {R} ^{n}}$
• The set of real number sequences with component-wise arrangement forms a Riesz space. ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$
• The set of real zero sequences with component-wise arrangement forms a Riesz space. ${\displaystyle c_{0}}$
• For is a Riesz room with a component arrangement. ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle l_{p}}$
• The set of bounded real sequences with a component-wise arrangement forms a Riesz space. ${\displaystyle l_{\infty }}$
• The set of continuous functions on an interval forms a Riesz space with pointwise arrangement. ${\displaystyle {\mathcal {C}}[a,b]}$ ${\displaystyle [a,b]}$
• The set of continuously differentiable functions on an interval forms an ordered vector space with the point-wise arrangement, but not a Riesz space. ${\displaystyle {\mathcal {C}}^{1}[a,b]}$ ${\displaystyle [a,b]}$

Integration theory

Riesz rooms offer prerequisites for an abstract theory of measure and integration. The central statement in this context is Freudenthal's spectral theorem . For Riesz spaces, this theorem guarantees the approximation property of functions through staircase functions in an abstract way . The Radon-Nikodym theorem and the Poisson summation formula for limited harmonic functions on the open disk are special cases of the Spectral Freudenthal. This spectral theorem was one of the starting points for the theory of Riesz spaces.

Individual evidence

1. ^ Riesz, Frigyes: Sur la décomposition des opérations fonctionelles linéaires , Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930) pp. 143-148

literature

• Luxemburg, WAJ & Zaanen, AC: "Riesz spaces", North-Holland, 1971, ISBN 978-0444866264
• VI Sobolev: Riesz space . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).