Should the Stein-Shakarchi Analysis Volumes be Studied in Order

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I couldn't find this question anywhere else, so just thought I would ask it here. There is a series of 4 analysis books written by Stein and Shakarchi (sometimes called the Princeton Lectures in Analysis), and as I intend to study from them, I was wondering whether it is important or not to learn the material in the order that the books were written:

  1. Fourier Analysis

  2. Complex Analysis

  3. Real Analysis

  4. Functional Analysis

For example, is the Fourier analysis in the first volume a prerequisite to understanding the complex analysis in the second volume? Or can one just proceed straight to complex analysis while having just a basic background in real analysis (like Tao or Rudin)? Thanks.

Best Answer

Most math books contain an introduction that outlines the prerequisites. I have only read Vol III, but that one was generally self-contained. I doubt an introductory complex analysis book would require Fourier analysis. Quoting from the intro to Vol III:

Despite the substantial connections that exist between the different volumes, enough overlapping material has been provided so that each of the first three books requires only minimal prerequisites: acquaintance with elementary topics in analysis such as limits, series, differentiable functions, and Riemann integration, together with some exposure to linear algebra.

In my opinion the chief prerequisite is "mathematical maturity", i.e., familiarity with proving stuff. They may not be so good as an introduction to proof-writing.