PREFACE These volumes deal with a subject, introduced half a century ago, that has become increasingly important and popular in recent years. While they cover the fundamental aspects of this subject, they make no attempt to be encyclopaedic. Their primary goal is to teach the subject and lead the reader to the point where the vast recent research literature, both in the subject proper and in its many applications, becomes accessible. Although we have put major emphasis on making the material pre- sented clear and understandable, the subject is not easy no account, however lucid, can make it so. If it is possible to browse in this subject and acquire a significant amount of information, we hope that these vol- umes present that opportunity—but they have been written primarily for the reader, either starting at the beginning or with enough preparation to enter at some intermediate stage, who works through the text systemati- cally. The study of this material is best approached with equal measures of patience and persistence. Our starting point in Chapter 1 is finite-dimensional linear algebra. We assume that the reader is familiar with the results of that subject and begin by proving the infinite-dimensional algebraic results that we need from time to time. These volumes deal almost exclusively with infinite- dimensional phenomena. Much of the intuition that the reader may have developed from contact with finite-dimensional algebra and geometry must be abandoned in this study. It will mislead as often as it guides. In its place, a new intuition about infinite-dimensional constructs must be culti- vated. Results that are apparent in finite dimensions may be false, or may be difficult and important principles whose application yields great re- wards, in the infinite-dimensional case. Almost as much as the subject matter of these volumes is infinite di- mensional, it is non-commutative real analysis. Despite this description, the reader will find a very large number of references to the "abelian" or "commutative" case—an important part of this first volume is an analysis of the abelian case. This case, parallel to function theory and measure theory, provides us with a major tool and an important guide to our Vll
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