# 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].

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