In the theory of linear topological spaces, the concept of a dual space is of paramount importance. In this short chapter I supplement this theory with a brief discussion of the dual spaces E∼ and E× which we have already seen, and which are fundamental to the ideas of the rest of the book. The most important new results concern the canonical evaluation map from a Riesz space E to the space of linear functionals on E× [32B below].

In the third section I give an equally abbreviated note on ‘perfect’ Riesz spaces, which include the majority of the spaces which interest us. Further general results concerning spaces in duality will be derived in the course of Chapter 8, while studying weak compactness.

The space E∼

This is the basic dual space associated with a given Riesz space E. Recall that so far we know that E∼ is the set of linear functionals on E which are bounded on order-bounded sets, and that it is a Dedekind complete Riesz space [16C, 16D]. The importance of the results below is that they apply to the evaluation map from E into G* for any solid linear subspace G of E∼; their principal application is of course in the case G = E×.