Riesz space

A Riesz space is a vector space with an association structure such that the linear and the association structure tolerated. In 1928, this area of Frigyes Riesz was defined and therefore today bears his name.

Definition

Be a vector space and a partially ordered set.

Then is called a Riesz space if the following axioms are satisfied:

Comments

First properties

For and following arithmetic rules apply:

• And
• And
• And
• Be looking for.

• And
• And

Examples

• The real numbers with the usual arrangement form a Riesz space.
• The one with componentwise arrangement forms a Riesz space.
• The set of real number sequences with component- wise arrangement forms a Riesz space.
• The set of real null sequences with component- wise arrangement forms a Riesz space.
• For with componentwise arrangement, a Riesz space.
• The set of bounded real sequences with component- wise arrangement forms a Riesz space.
• The set of continuous functions on an interval forms with point- wise arrangement of a Riesz space.
• The amount of continuously differentiable functions on an interval is an ordered vector space with the pointwise order, but no Riesz space.

Integration theory

Riesz spaces provide conditions for an abstract measure and integration theory. The central message in this context is the spectral theorem of Freudenthal. This set guarantees for Riesz spaces in an abstract way the approximation properties of functions by step functions. The Radon - Nikodym theorem and the Poisson summation formula for bounded harmonic functions on the open disk are special cases of the Spectral Theorem of Freudenthal. This spectral theorem was one of the starting points for the theory of Riesz spaces.