∞ Real Analysis
Based on Jiří Lebl, "Basic Analysis I", licensed CC BY-SA 4.0.
The rigorous foundation beneath calculus: limits, continuity, convergence, and integration, built from the axioms of the real numbers. Translated into runnable code with diagrams.
| Chapter | |||
|---|---|---|---|
| 1. | Real Numbers | The axioms, completeness, supremum, and why the rationals aren't enough | ∞ |
| 2. | Sequences | Convergence, monotone sequences, Bolzano-Weierstrass, and Cauchy completeness | ∞ |
| 3. | Series | When infinite sums converge: comparison, ratio, root tests, and power series | ∞ |
| 4. | Continuous Functions | Epsilon-delta continuity, IVT, EVT, and uniform continuity | ∞ |
| 5. | The Derivative | Definition, chain rule, mean value theorem, L'Hopital, and Taylor's theorem | ∞ |
| 6. | The Riemann Integral | Partitions, upper/lower sums, integrability, and the fundamental theorem | ∞ |
| 7. | Sequences of Functions | Pointwise vs uniform convergence, interchange of limits, Weierstrass M-test | ∞ |
| 8. | Metric Spaces | Open/closed sets, compactness, connectedness, and completeness in the abstract | ∞ |
📺 Video lectures: MIT 18.100B Real Analysis
Neighbors
- ∫ Calculus — real analysis proves what calculus does
- ✎ Proofs — epsilon-delta arguments are the proof technique here
- 🎰 Probability — measure theory extends real analysis to probability
- △ Geometry — metric spaces appear in both real analysis and geometry