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 Андрей Николаевич Колмогоров (Andrey Nikolaevich Kolmogorov) and Сергей Васильевич Фомни (Sergei Vasilyevich Fomin).

The original text in 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.


As no official errata page is available, I will collate errors that Jeffrey Kwan and I come across below. Charlie Hoang has also uploaded a collection of errata here, but it contains some inaccuracies (for example the third supposed error is actually correct as it is originally, since the solution for Problem 1 on page 76 is to show that completeness, not just the inequality, is required).

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

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

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

  • 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 \(P(a)\) 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

  • Page 56: Definition 2. A sequence, not a subsequence.

  • Page 64: Twice: \([R] = R^*\), not \([R] = R\).

  • Page 72: In the statement of Theorem 2’: \(\tilde{y}_n\), not \(_n \tilde{y}\).

  • Page 85: Definition 4. \(x \notin O_y\) and \(y \notin O_x\), not \(x \in O_y\) and \(y \in O_x\).

  • Page 95: Theorem 7 proof. \(F_n = \{x_n, x_{n+1}, \ldots\}\), not \(x_n = \{x_n, x_{n+1}, \ldots\}\).

  • Page 98: Example 2 should come after Definition 2.

  • Page 100: Theorem 2 proof. finitely many of them, not infinitely many of them.

  • Page 105: Theorem 5 proof, equation (3). \(\phi^{(n)}(x’)\), not \(\phi(x’)\).

  • Page 107: Problem 5. \(y_j\), not \(y_i\).

On the display of mathematical symbols

Typically websites call upon MathJax to wrangle \(\LaTeX\) incantations into a more standard and readable form. MathML is a JavaScript-free alternative. However, until I am able to figure out how to export org-mode to HTML using latexmlc or latex2mathml without causing Emacs to stutter, I apologise for the use of JavaScript on an otherwise script-free site.

Sketched notes

My notes are simply for the purposes of jogging my memory. CoveredInChocolate also keeps notes which can be found here.

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

    1. Sets and Functions

    2. Equivalence of Sets. The Power of a Set

    3. Ordered Sets and Ordinal Numbers

    4. Systems of Sets

  2. Metric Spaces

    1. Basic Concepts

    2. Convergence. Open and Closed Sets

    3. Complete Metric Spaces

    4. Contraction Mappings

  3. Topological Spaces

    1. Basic Concepts

    2. Compactness

    3. Compactness in Metric Spaces

In progress.