Mathematical Analysis
by 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.

See also