83 They cannot scare me with their empty spaces Between stars—on stars where no human race is. I have it in me so much nearer home To scare myself with my own desert places. —Robert Frost 3 Space Spatial Feeling William James, known among religion scholars for his ame- nable approach to the subject of religion in The Varieties of Reli- gious Experience (1902), was deeply interested in the matter of space and spatial perception throughout his writing career. It was the topic of his first published article, “The Spatial Quale” (1879), and it comprised the

. And that movie watching had nothing to do with space or history or men or monkeys. All he wanted was sex. I’d waited a couple of years for that: to have seen him broken and mended and then looking right through me, right on to the stars. I would still want to mean so little that I’d be see-thru. Sometimes I can daydream him back to breakable. Mostly, I would want to watch movies with him while he’s in love with someone else and helping himself to me.

41 Topological Vector Spaces We now consider our second of two examples of structures that mix topology and algebra. Topological vector spaces are somewhat more useful, a bit more complicated, and considerably richer in structure than topological groups. We regard the previous chapter as just an introduction to the present one. " 11" Let V be a real vector space. We write V X V -> V, R X V -> V, and t V --+ V for the mappings with action u( v, v') = v + v', 11"{ a, v) = av, and t( v) = -v, respectively. A (real) topological vector space consists of four

44 Constructing Measure Spaces In this chapter, we describe various techniques that yield measure spaces from measure spaces. p " Let X, M, M -+ R be a measure space. Let K be a measurable subset of X Denote by N the collection of all subsets of K which, as subsets of X, are measurable (Le., which are in M). For A in N, set v(A) = Il(A). Then K, /I N, N -+ R" is a measure space. [Proof: The first, third, and fourth conditions for a measure space are immediate from those conditions on measure space X For the second condition, note that, for A in N, the

38 Uniform Spaces Let X be a set. We may regard a topology on this set as providing one with the following structure: the notion of what are the neighborhoods of each point of X On the other hand, a metric (chapter 26) on X provides some- what more structure: not only can one speak of the neighborhoods of each point of X, but also one can assign to these neighborhoods a "numerical size." (For example, a neighborhood of "radius 7" about point x consists of all x' with d(x,x') < 7.) In particular, one can, in a metric space, "compare sizes of different

48 Hilbert Spaces A Hilbert space consists of two things-i) a complex vector space H, and ii) a rule that assigns, to any two vectors hand h' in H, a complex number (writ- ten (h,h') and called the inner product of hand h')-subject to the following four conditions: 1. For any vectors h, h', and h" , and any complex number c, we have (h + ch' ,h") = (h,h") + C(h' ,h") and (h,h' + ch") = (h,h') + c(h,h") , where a bar over a complex number denotes the complex-conjugate of that number. 2. For any vectors hand h' , we have (h,h') = (h',h) . 3. For any

43 Measure Spaces We now begin our discussion of the theory of integration. We proceed in two steps. In the first, we introduce the notion of measure, which appropriately generalizes that of "area" (e.g., of regions in the plane). In the second, we use this measure to define the notion of an integral. We first introduce a little notation. Denote by R* the set consisting of the non-negative real numbers, together with one additional element, which we write 00. The element 00 of R* is called infinite, the others finite. For a and b elements of R*, we write

26 Topological Spaces We now begin the study of topological spaces. As we shall see, the viewpoint is somewhat different in topology than it was in the study of groups, vector spaces, etc. If one had to characterize this difference in terms of a single feature, perhaps it would be this: whereas in the algebraic categories one is concerned principally with elements (of sets), one is in topology more con- cerned with subsets. Nonetheless, a number of the ideas used in the algebraic categories have topological versions. A topological space consists of two