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Manual Modern real analysis

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Table of Contents 1. Real, Cardinal, and Ordinal Numbers. Elements of Topology.

Spaces: An Introduction to Real Analysis

Measure Theory. Measurable Functions.

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Elements of Functional Analysis. Measures and Linear Functionals. Functions of Several Variables. The first part of the book, Elements of Real Analysis, addresses single-variable calculus — limits, continuity, derivatives, integrals, sequences, and series. The second part, Elements of Abstract Analysis, covers more advanced topics — for example, Banach spaces, the Daniell integral, the Baire Category Theorem, and integral operators.

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This axiomatic approach allows one to work with a hyperreal system without having to deal with its explicit construction in terms of ultrapowers and the like. However, Vakil actually presents the ultrapower construction first and then uses it as motivation for internal set theory.


  1. William P Ziemer (Author of Modern Real Analysis);
  2. Bibliographic Information;
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Real Analysis through Modern Infinitesimals contains a large four-digit number of exercises, both incorporated into the text and gathered into separate sections. Answers are not included.

In recent years Leon Harkleroad has mostly concentrated on mathematical aspects of music, but he still enjoys revisiting his old stomping grounds of mathematical logic. Preface Introduction Part I.

Elements of Real Analysis: 1. Internal set theory 2.

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The real number system 3. Sequences and series 4. The topology of R 5. Limits and continuity 6.

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Real Analysis

Tue Convolutions: definition, approximations of the identity 3. Convolutions: convergence, smoothing, Poisson kernel 3. Differentiation of measures 4.

Riesz representation theorem and partitions of unity 5. Weak convergence and compactness of Radon measures 5.

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Further reading A. Main course page Assignments Contact information. MS-E - Real analysis, Course home page.