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## Main Question or Discussion Point

**Author:**Rudin**Title:**Principles of Mathematical Analysis**Amazon Link:**https://www.amazon.com/dp/007054235X/?tag=pfamazon01-20**Prerequisities:**Rigorous calculus, including epsilon-delta proofs. Spivak's "Calculus" would be more than sufficient preparation.

**Table of Contents:**

Code:

```
[LIST]
[*] Preface
[*] The Real and Complex Number Systems
[LIST]
[*] Introduction
[*] Ordered Sets
[*] Fields
[*] The Real Field
[*] The Extended Real Number System
[*] The Complex Field
[*] Euclidean Spaces
[*] Appendix
[*] Exercises
[/LIST]
[*] Basic Topology
[LIST]
[*] Finite, Countable, and Uncountable Sets
[*] Metric Spaces
[*] Compact Sets
[*] Perfect Sets
[*] Connected Sets
[*] Exercises
[/LIST]
[*] Numerical Sequences and Series
[LIST]
[*] Convergent Sequences
[*] Subsequences
[*] Cauchy Sequences
[*] Upper and Lower Limits
[*] Some Special Sequences
[*] Series
[*] Series of Nonnegative Terms
[*] The Number e
[*] The Root and Ratio Tests
[*] Power Series
[*] Summation by Parts
[*] Absolute Convergence
[*] Addition and Multiplication of Series
[*] Rearrangements
[*] Exercises
[/LIST]
[*] Continuity
[LIST]
[*] Limits of Functions
[*] Continuous Functions
[*] Continuity and Compactness
[*] Continuity and Connectedness
[*] Discontinuities
[*] Monotonic Functions
[*] Infinite Limits and Limits at Infinity
[*] Exercises
[/LIST]
[*] Differentiation
[LIST]
[*] The Derivative of a Real Function
[*] Mean Value Theorems
[*] The Continuity of Derivatives
[*] L'Hospital's Rule
[*] Derivatives of Higher Order
[*] Taylor's Theorem
[*] Differentiation of Vector-valued Functions
[*] Exercises
[/LIST]
[*] The Riemann-Stieltjes Integral
[LIST]
[*] Definition and Existence of the Integral
[*] Properties of the Integral
[*] Integration and Differentiation
[*] Integration of Vector-valued Functions
[*] Rectifiable Curves
[*] Exercises
[/LIST]
[*] Sequences and Series of Functions.
[LIST]
[*] Discussion of Main Problem
[*] Uniform Convergence
[*] Uniform Convergence and Continuity
[*] Uniform Convergence and Integration
[*] Uniform Convergence and Differentiation
[*] Equicontinuous Families of Functions
[*] The Stone-Weierstrass Theorem
[*] Exercises
[/LIST]
[*] Some Special Functions
[LIST]
[*] Power Series
[*] The Exponential and Logarithmic Functions
[*] The Trigonometric Functions
[*] The Algebraic Completeness of the Complex Field
[*] Fourier Series
[*] The Gamma Function
[*] Exercises
[/LIST]
[*] Functions of Several Variables
[LIST]
[*] Linear Transformations
[*] Differentiation
[*] The Contraction Principle
[*] The Inverse Function Theorem
[*] The Implicit Function Theorem
[*] The Rank Theorem
[*] Determinants
[*] Derivatives of Higher Order
[*] Differentiation of Integrals
[*] Exercises
[/LIST]
[*] Integration of Differential Forms
[LIST]
[*] Integration
[*] Primitive Mappings
[*] Partitions of Unity
[*] Change of Variables
[*] Differential Forms
[*] Simplexes and Chains
[*] Stokes' Theorem
[*] Closed Forms and Exact Forms
[*] Vector Analysis
[*] Exercises
[/LIST]
[*] The Lebesgue Theory
[LIST]
[*] Set Functions
[*] Construction of the Lebesgue Measure
[*] Measure Spaces
[*] Measurable Functions
[*] Simple Functions
[*] Integration
[*] Comparison with the Riemann Integral
[*] Integration of Complex Functions
[*] Functions of Class [itex]\mathcal{L}^2[/itex]
[*] Exercises
[/LIST]
[*] Bibliography
[*] List of Special Symbols
[*] Index
[/LIST]
```

**User comments:**

- jbunniii

For the well prepared reader, this is a beautifully clear treatment of the main topics of undergraduate real analysis. Yes, it is terse. Yes, the proofs are often slick and require the reader to fill in some nontrivial gaps. No, it doesn't spend much time motivating the concepts. It is not the best book for a first exposure to real analysis - that honor belongs to Spivak's "Calculus." But don't kid yourself that you have really mastered undergraduate analysis if you can't read Rudin and appreciate its elegance. It also serves as a nice, clean, uncluttered reference which few graduate students would regret having on their shelves.

- micromass

This is a wonderful book**iff**you can handle it. Do not use Rudin as your first exposure to analysis, it will be a horrible experience. However, if you already completed a Spivak level text, then Rudin will be a wonderful experience. It contains many gems and many challenging problems. Personally, I find his approach to differential forms and Lebesgue integration quite weird though. I think there are many books that cover it better than him. But the rest of the book is extremely elegant and nice.

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