Chapter 1. Preliminaries
Chapter 2. Topological spaces
Chapter 3. A first look at Hilbert spaces and Fourier series
Chapter 4. Measure theory
Chapter 5. Convexity and Banach spaces
Chapter 6. Tempered distributions and the Fourier transform
Chapter 7. Bonus chapter: Probability basics
Chapter 8. Bonus chapter: Hausdorff measure and dimension
Chapter 9. Bonus chapter: Inductive limits and ordinary distributions

Part 1 is devoted to real analysis. From one point of view, it presents the infinitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribution theory). From another, it shows the triumph of abstract spaces: topological spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally convex spaces, Fréchet spaces, Schwartz space, and Lp spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fixed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable functions, Brownian motion, space-filling curves, solutions of the moment problem, Haar measure, and equilibrium measures in potential theory. --- summary provided by publisher