Analysis is the infinitesimal calculus writ large. Calculus as taught tomost high school students and college freshmen is the subject as it existed about 1750—I’ve no doubt that Euler could have gotten a perfect score on the Calculus BC advanced placement exam. Even “rigorous” calculus courses that talk about ε-δ proofs and the intermediate value theorem only bring the subject up to about 1890 after the impact of Cauchy and Weierstrass on real variable calculus was felt.
This volume can be thought of as the infinitesimal calculus of the twentieth century. From that point of view, the key chapters are Chapter 4, which covers measure theory—the consummate integral calculus—and the first part of Chapter 6 on distribution theory—the ultimate differential calculus. But from another point of view, this volume is about the triumph of abstraction. Abstraction is such a central part of modern mathematics that one forgets that it wasn’t until Fr´echet’s 1906 thesis that sets of points with no a priori underlying structure (not assumed points in or functions on Rn) are considered and given a structure a posteriori (Fr´echet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology, and logic.
Abstract spaces are a distinct thread here, starting with topological spaces in Chapter 2, Banach spaces in Chapter 5 (and its special case, Hilbert spaces, in Chapter 3), and locally convex spaces in the later parts of Chapters 5 and 6 and in Chapter 9.
Of course, abstract spaces occur to set up the language we need for measure theory (which we do initially on compact Hausdorff spaces and where we use Banach lattices as a tool) and for distributions which are defined as duals of some locally convex spaces. Besides the main threads of measure theory, distributions, and abstract spaces, several leitmotifs can be seen: Fourier analysis (Sections 3.5, 6.2, and 6.4–6.6 are a minicourse), probability (Bonus Chapter 7 has the basics, but it is implicit in much of the basic measure theory), convexity (a key notion in Chapter 5), and at least bits and pieces of the theory of ordinary and partial differential equations.
The role of pivotal figures in real analysis is somewhat different from complex analysis, where three figures—Cauchy, Riemann, andWeierstrass— dominated not only in introducing the key concepts, but many of the most significant theorems. Of course, Lebesgue and Schwartz invented measure theory and distributions, respectively, but after ten years, Lebesgue moved on mainly to pedagogy and H¨ormander did much more to cement the theory of distributions than Schwartz. On the abstract side, F. Riesz was a key figure for the 30 years following 1906, with important results well into his fifties, but he doesn’t rise to the dominance of the complex analytic three. In understanding one part of the rather distinct tone of some of this volume, the reader needs to bear in mind “Simon’s three kvetches”:2
1. Every interesting topological space is a metric space.
2. Every interesting Banach space is separable.
3. Every interesting real-valued function is Baire/Borel measurable.