Introductory Real Analysis

Introductory Real Analysis is a translation, and revision, by Richard A. Silverman of the second edition of Элементы теории функций и функционального анализа (Elements of the Theory of Functions and Functional Analysis), written by Андре́й Никола́евич Колмого́ров (Andrey Nikolaevich Kolmogorov) and Серге́й Васи́льевич Фоми́н (Sergei Vasilyevich Fomin). The original Russian ran to a total of four editions. Based off of a cursory glance, Silverman’s translation is more fluent (and up to date) than the overly literal translation by Leo F. Boron of the first edition called Elements of the Theory of the Theory of Functions and Functional Analysis. However, Silverman appears to have introduced several mathematical errors.

The prose and diagrammes are terse, clear, and enlightening. The problems within bite. What more could a budding mathematician hope for?

Errata

As (seemingly) no official errata page is available (though a collection uploaded by Charlie Hoang is available here), I will collate errors Jeffrey Kwan and I come across below:

  • Page 11: 0 1 = 0 0 1 0 \dfrac{0}{1}=0 has a rational number height of 1 1 1 , not 0 0 .

  • Page 17: On the other hand… should discuss if x X 𝑥 𝑋 x\in X , then x X 𝑥 𝑋 x\notin X , instead of repeating the same scenario twice.

  • Page 23: Definition 1. μ 𝜇 \mu such that μ a 𝜇 𝑎 \mu\leq a not μ < a 𝜇 𝑎 \mu<a .

  • Page 27: It follows from the well-ordering theorem and Theorem 4, not Theorem 5.

  • Page 28, 29: There are two Theorem 4s. The second Theorem 4 (mathematical induction) should be Theorem 5, and accordingly the transfinite induction theorem should be renamed Theorem 5’.

  • Page 29: Suppose P ( a ) 𝑃 𝑎 P(a) fails to be true for, not tor.

  • Page 30: Problem 4. By a lattice is meant a partially ordered set which has both a greatest

Sketched notes

My notes are simply for the purposes of jogging my memory. CoveredInChocolate has more substantial notes here. Nalin Pithwa also has notes here, although Jeffrey has spotted some errors in a couple of Pithwa’s solutions.

My solutions for problems may simply be sketched out too, especially in cases where the LaTeX is particularly cumbersome. Any corrections, or clarification of open ends, would be greatly welcomed.

  1. Set theory