by Gordon H. Fullerton

- Metric and normed spaces
- Open and closed sets
- Compactness
- Connectedness
- Convergence
- Consequences of completeness

Problems 1Chapter 2. CONTINUOUS FUNCTIONS

- Definition and topological conditions
- Preservation of compactness and connectedness
- Uniform convergence
- Uniform continuity
- Weierstrass's Theorem
- The Stone-Weierstrass Theorem
- Compactness in
*C(X)* - Topological spaces: an aside

Problems 2Chapter 3. FURTHER RESULTS ON UNIFORM CONVERGENCE

- Uniform convergence and integration
- Uniform convergence and differentiation
- Uniform convergence of series
- Tests for uniform convergence of series
- Power series

Problems 3Chapter 4. LEBESGUE INTEGRATION

- The collection
*K*and null sets - The Lebesgue integral
- Convergence theorems
- Relation between Riemann and Lebesgue integration
- Daniell integrals
- Measurable functions and sets
- Complex-valued functions: L^p spaces
- Double integrals

Problems 4Chapter 5. FOURIER TRANSFORMS

- L^1 theory: elementary results
- The inversion theorem
- L^2 theory

Problems 5

Chapter 1. METRIC SPACES

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.

Topological Spaces: From Distance to Neighborhood by Gerard Buskes and Arnoud van Rooij

Analiza matematyczna: funkcje jednej zmiennej by Tadeusz Krasinski (book in Polish)