Here are my notes for real analysis based on Chapters 1 to 7 of Rudin’s Principles of Mathematical Analysis. We start from the very beginning: defining what a “real number” is. Then, we introduce metric spaces and study their topological properties. With this background, we proceed to give a rigorous treatment of everything from “first-year” calculus: sequences and series, limits and continuity, differentiation, and integration. We finish with a chapter on uniform convergence, which is all about interchanging limits.

In addition to the standard definitions, theorems, and proofs, there are lots of exercises that I hope will challenge the reader.

Prerequisites:

  • First-year calculus: limits, continuity, differentiation, integration, and series convergence tests.
  • Basic proof techniques: direct proof, proof by contrapositive, proof by contradiction, proof by (weak or strong) induction.
  • Basic set theory: intersections, unions, set complements, and how they interact with each other (e.g. you should know that $A\cap (B\cup C) = (A\cap B)\cup (A\cap C)$).
  • Linear algebra: linear transformations, null space and column space, determinants, eigenvalues and eigenvectors, orthogonality and projections.

Download the PDF notes here