This chapter and the next are an exposition of the elementary theory of Riesz spaces with linear space topologies on them; they form a natural confluence of the Riesz space theory of Chapter 1 with ordinary abstract functional analysis. The present chapter proceeds by studying a series of properties that Riesz space topologies can have (most of them being, of course, relations between the topology and the order structure). The properties are all chosen to be ones possessed by important special cases; the theory never gets far from the applications that will be made of it, except perhaps in the work leading up to Nakano's theorem in §23. In §26 the conditions imposed are so strong that they become significantly less abstract; this section almost forms a layer intermediate between the first five sections of this chapter and the concrete examples from measure theory that will follow later.

Throughout this chapter I avoid the hypothesis of local convexity as far as possible. This is for both positive and negative reasons. The negative justification is that it can be done without an excessive amount of extra work. On the positive side, the most important examples of non-locally-convex spaces in analysis are Riesz spaces and are particularly accessible by the methods of this chapter [see, for example, 63K]. But of course it is quite possible to read this chapter with only locally convex spaces in mind.