| Topic Number | Topic | Text Section | Subtopics |
| 1 |
Euclidean Spaces Rn |
N/A | Linear spaces |
| 8.2 | Algebraic structure |
| 8.2 | Planes and linear transformations |
| 8.3 | The Topology of Rn |
| 8.4 | Interior, closure, and boundary |
| 2 |
Metric Spaces |
10.1 | Introduction and definitions |
| 10.2 | Limits of functions |
| 10.3 | Interior, closure, and boundary |
| 10.4 | Compactness |
| 10.5 | Connectedness |
| 10.6 | Continuity |
| 3 |
Series |
N/A | The meaning of infinite sums; rearrangements |
| 6.1 | Convergence of series |
| 6.2 | Series with noonnegative terms |
| 6.3 | Absolute and conditional convergence |
| 6.4 | Alternating series |
| 6.5 | Estimation and truncation error |
| N/A | Double summation |
| N/A | Products of infinite series |
| 4 |
Sequences and Series of Functions |
7.1 | Pointwise and uniform convergence of sequences |
| 7.2 | Pointwise, absolute, and uniform convergence of series |
| N/A | Uniform convergence, differentiability, and integrability |
| N/A | Equicontinuous families and the Ascoli-Arzela theorem |
| 7.3 | Power series |
| 7.4 | Real analytic functions |
| 10.7 | The Stone-Weierstrass Theorem |
| 5 |
Fourier Series |
14.1 | Definitions |
| 14.2 | Summability |
| 14.3 | Growth of coefficients |
| 14.4 | Convergence |
| 14.5 | Uniqueness |
| 6 |
Lebesgue Measure and Integration |
N/A | Algebras of sets; set functions and additivity |
| N/A | Measure and measurable functions |
| N/A | The Lebesgue integral |