I've had no interest in the complex half, but I can claim to have done well enough with the real half. I like it. He does especially well with Fourier things,

I've had no interest in the complex half, but I can claim to have done well enough with the real half. I like it. He does especially well with Fourier things, which is not surprising considering some of his other work, He gives the von Neumann proof of the Radon-Nikodym theorem. I prefer the approaches with more steps with each step smaller, but it's great to see the von Neumann masterstroke. Rudin's chapter on Banach space knocks off the main results quickly and cleanly. His construction of Lebesgue measure early on is a bit long but surprising and, net, shorter than the other ways to do it. Curiously he has a nice chapter with more general results on change of variable in multiple integration. He does well with the various duality results, which are surprising and important. A good competitor, of course, is Royden, 'Real Analysis'.