# Riesz Spaces

• Michel Simonnet
Part of the Universitext book series (UTX)

## Summary

1.1 This first section introduces the notion of ordered groups and lattices. It might be useful to keep in mind some well known examples, such as Z or R or the space C([a, b]) of real-valued continuous functions on [a, b] with their natural orderings.

1.2 It is devoted to a short study of Riesz spaces, that is, ordered vector spaces which are lattices. The space X of real-valued continuous functions on [a, b], endowed with its usual ordering, is an example of Riesz space.

1.3 In this section we turn our attention to the duals of Riesz spaces. If E is any ordered vector space, there is a natural notion of positive linear form: a linear form I is said to be positive if the image of any positive element of E under I is a positive real number. Notice that if E is the space of real-valued continuous functions on a closed interval [a, b], equipped with its natural ordering, a positive linear form on E is continuous.

1.4 We finally define Daniell measures. A Daniell measure is a linear form L on a space of functions H which has a “finite variation” (Definition 1.4.1) and such that L(f n ) → 0 whenever f n decreases to 0 in H. This last condition should be understood as a continuity condition. Among the examples that the reader may want to keep in mind axe the Riemann integral considered as a positive linear form on the set of continuous functions on [a, b] and also the positive linear forms ε c : Xff(c) where c ∈ [a, b].

## Keywords

Boolean Algebra Nonempty Subset Positive Element Vector Subspace Riesz Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.