What topics are covered in Real Analysis by Carothers?

[micromass]

• Author: Carothers
• Title: Real Analysis
• Amazon link https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous calculus.
• Level: Grad

Table of Contents:

[LIST]
[*] Preface
[*] Metric Spaces
[LIST]
[*] Calculus Review
[LIST]
[*] The Real Numbers
[*] Limits and Continuity
[*] Notes and Remarks
[/LIST]
[*] Countable and Uncountable Sets
[LIST]
[*] Equivalence and Cardinality
[*] The Cantor Set
[*] Monotone Functions 
[*] Notes and Remarks
[/LIST]
[*] Metrics and Norms
[LIST]
[*] Metric Spaces
[*] Normed Vector Spaces
[*] More Inequalities
[*] Limits in Metric Spaces
[*] Notes and Remarks
[/LIST]
[*] Open Sets and Closed Sets
[LIST]
[*] Open Sets
[*] Closed Sets
[*] The Relative Metric
[*] Notes and Remarks
[/LIST]
[*] Continuity
[LIST]
[*] Continuous Functions
[*] Homeomorphisms
[*] The Space of Continuous Functions 
[*] Notes and Remarks
[/LIST]
[*] Connectedness
[LIST]
[*] Connected Sets
[*] Notes and Remarks
[/LIST]
[*] Completeness
[LIST]
[*] Totally Bounded Sets
[*] Complete Metric Spaces
[*] Fixed Points
[*] Completions
[*] Notes and Remarks
[/LIST]
[*] Compactness
[LIST]
[*] Compact Metric Spaces
[*] Uniform Continuity
[*] Equivalent Metrics
[*] Notes and Remarks
[/LIST]
[*] Category
[LIST]
[*] Discontinuous Functions
[*] The Baire Category Theorem
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] Function Spaces
[LIST]
[*] Sequences of Functions
[LIST]
[*] Historical Background
[*] Pointwise and Uniform Convergence 
[*] Interchanging Limits
[*] The Space of Bounded Functions
[*] Notes and Remarks
[/LIST]
[*] The Space of Continuous Functions
[LIST]
[*] The Weierstrass Theorem
[*] Trigonometric Polynomials
[*] Infinitely Differentiable Functions
[*] Equicontinuity
[*] Continuity and Category
[*] Notes and Remarks
[/LIST]
[*] The Stone-Weierstrass Theorem
[LIST]
[*] Algebras and Lattices
[*] The Stone-Weierstrass Theorem
[*] Notes and Remarks
[/LIST]
[*] Functions of Bounded Variation
[LIST]
[*] Functions of Bounded Variation
[*] Helly's First Theorem
[*] Notes and Remarks
[/LIST]
[*] The Riemann-Stieltjes Integral
[LIST]
[*] Weights and Measures
[*] The Riemann-Stieltjes Integral
[*] The Space of Integrable Functions
[*] Integrators of Bounded Variation
[*] The Riemann Integral
[*] The Riesz Representation Theorem
[*] Other Definitions, Other Properties
[*] Notes and Remarks
[/LIST]
[*] Fourier Series
[LIST]
[*] Preliminaries
[*] Dirichlet's Formula
[*] Fejer's Theorem
[*] Complex Fourier Series
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] Lebesgue Measure and Integration
[LIST] 
[*] Lebesgue Measure
[LIST]
[*] The Problem of Measure
[*] Lebesgue Outer Measure
[*] Riemann Integrability
[*] Measurable Sets
[*] The Structure of Measurable Sets
[*] A Nonmeasurable Set 
[*] Other Definitions
[*] Notes and Remarks
[/LIST]
[*] Measurable Functions
[LIST]
[*] Measurable Functions
[*] Extended Real-Valued Functions
[*] Sequences of Measurable Functions
[*] Approximation of Measurable Functions
[*] Notes and Remarks
[/LIST]
[*] The Lebesgue Integral
[LIST]
[*] Simple Functions
[*] Nonnegative Functions
[*] The General Case
[*] Lebesgue's Dominated Convergence Theorem
[*] Approximation of Integrable Functions
[*] Notes and Remarks
[/LIST]
[*] Additional Topics 
[LIST]
[*] Convergence in Measure
[*] The [itex]L_p[/itex] Spaces
[*] Approximation of [itex]L_p[/itex] Functions 
[*] More on Fourier Series
[*] Notes and Remarks
[/LIST]
[*] Differentiation
[LIST]
[*] Lebesgue's Differentiation Theorem
[*] Absolute Continuity
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] References
[*] Symbol Index
[*] Topic Index 
[/LIST]

 

[jbunniii]

This is a very nice book on metric space topology, function spaces, and integration (both Riemann-Stieltjes and Lebesgue). It is exceptionally well written and is at about the same level of sophistication as Rudin's Principles of Mathematical Analysis, without being so terse and austere. Indeed, this book has quite a lively and detailed discussion, providing a great deal of motivation that is largely absent from Rudin. The proofs are also more detailed. The emphasis is very much on the three topics I listed above: it doesn't contain anything about differentiation, power series, and other standard topics, so this cannot be one's only real analysis book. But for what it does cover, it's excellent.

 

[lugita15]

I think I'd disagree with it being a graduate-level book. I think any undergraduate who's seen epsilon-delta before should be able to easily handle it.

I love how Carothers intersperses exercises throughout the chapter, to guide the student in learning the material. It may be the best introduction to real analysis I've seen.

 

[WannabeNewton]

Let's not forget about the awesome historical accounts and the insane amount of exercises. Carothers has a truly poetic way of writing.

 

[lugita15]

Yes, the historical accounts are another great feature of the book.