Real Analysis (2005), Real Analysis and Applications (2005) - Frank Morgan, American Mathematical Society

I was introduced to these books when my son David was about to take a class in variational calculus from my friend and colleague, Fred Hickling.  Fred mentioned that Morgan's books were well though out pedagogically.  The calculus of variations is in Real Analysis and Applications and is being used as prelude to Gelfand and Fomin.  If you are a physicist and want to get acquaint yourself with modern mathematical notation - necessary if you want to read Arnol'd's Mathematical Methods of Classical Mechanics for instance - these books are a good choice.  You don't need both as there is a considerable overlap in content, but either would be a good addition to the mathematical library of a physicist.  I still find set theoretic notation repellent, but these books make it easier to review.

Each of the books contain short chapters that summarize information and outline proofs. The chapters in Real Analysis are:

Part I: Real numbers and limits

1. Numbers and logic
2.   Infinity
3.   Sequences
4.   Functions and limits

Part II: Topology

5. Open and closed sets
6.   Continuous functions
7.   Composition of functions
8.   Subsequences
9.  Compactness
10. Existence of maximum
11. Uniform continuity
12. Connected sets and the intermediate value theorem
13. The Cantor set and fractals

Part III: Calculus

14. The derivative and the mean value theorem
15. The Riemann integral
16. The fundamental theorem of calculus
17. Sequences of functions
18. The Lebesgue theory
19. Infinite series ∑an
20. Absolute convergence
21. Power series
22. Fourier series
23. Strings and springs
24. Convergence of Fourier series
25. The exponential function
26. Volumes of n-balls and the gamma function

 Part IV: Metric spaces

27. Metric spaces
28. Analysis on metric spaces
29. Compactness in metric spaces
30. Ascoli's theorem

The chapters in Real Analysis and Applications are:

Part I: Real numbers and limits

1.  Numbers and logic
2.  Infinity
3.  Sequences
4.  Subsequences
5.  Functions and limits
6.  Composition of functions

Part II: Topology

7.  Open and closed sets
8.  Compactness
9.  Existence of maximum
10. Uniform continuity
11. Connected sets and the intermediate value theorem
12.The Cantor set and fractals

Part III: Calculus

13. The derivative and the mean value theorem
14. The Riemann integral
15. The fundamental theorem of calculus
16. Sequences of functions
17.The Lebesgue theory
18. Infinite series
19. Absolute convergence
20. Power series
21. The exponential function
22. Volumes of n-balls and the gamma function

Part IV: Fourier series

23. Fourier series
24. Strings and springs
25. Convergence of Fourier series

Part V: The calculus of variations

26. Euler's equation
27. First integrals and the Brachistochrone problem
28. Geodesics and great circles
29. Variational notation, higher order equations
30, Harmonic functions
31. Minimal surfaces
32. Hamilton's action and Lagrange's equations
33. Optimal economic strategies
34. Utility of consumption
35. Riemannian geometry
36. Noneuclidean geometry
37. General relativity

Both books contain partial solutions to problems.  Real Analysis has 155 numbered pages. Real Analysis and Applications has 197 numbered pages.  Either is worth owning, I have a slight preference for the longer book.