# Mathematical Analysisby Gordon H. Fullerton

Chapter 1. METRIC SPACES

1. Metric and normed spaces
2. Open and closed sets
3. Compactness
4. Connectedness
5. Convergence
6. Consequences of completeness
Problems 1

Chapter 2. CONTINUOUS FUNCTIONS

7. Definition and topological conditions
8. Preservation of compactness and connectedness
9. Uniform convergence
10. Uniform continuity
11. Weierstrass's Theorem
12. The Stone-Weierstrass Theorem
13. Compactness in C(X)
14. Topological spaces: an aside
Problems 2

Chapter 3. FURTHER RESULTS ON UNIFORM CONVERGENCE

15. Uniform convergence and integration
16. Uniform convergence and differentiation
17. Uniform convergence of series
18. Tests for uniform convergence of series
19. Power series
Problems 3

Chapter 4. LEBESGUE INTEGRATION

20. The collection K and null sets
21. The Lebesgue integral
22. Convergence theorems
23. Relation between Riemann and Lebesgue integration
24. Daniell integrals
25. Measurable functions and sets
26. Complex-valued functions: L^p spaces
27. Double integrals
Problems 4

Chapter 5. FOURIER TRANSFORMS

28. L^1 theory: elementary results
29. The inversion theorem
30. L^2 theory
Problems 5

Excerpt from the Preface:

In Chapter 1, the concepts of compactness, connectedness and completeness are introduced in a metric space setting. Chapter 2 is concerned with properties of continuous functions between metric spaces. Uniform convergence of sequences of functions is defined and related to the completeness of C(X). Proofs of the Stone-Weierstrass and Ascoli theorems are given. The possibility of extending much of the above material to a general topological setting is briefly indicated. Chapter 3 deals with further results on the uniform convergence sequences and series of functions. In Chapter 4, the Lebesgue integral is defined, using Daniell's approach, as a linear functional on a certain class of functions. The relation to the traditional measure-theoretic definition is clearly explained. A discussion of the L^p spaces and double integrals is included. Chapter 5 contains the basic results of the L^1 and L^2 theories of Fourier transforms.