Chapter 1
Section 1.1: Proofs and logic: Video. Notes.
Section 1.2: Sets: Video. Notes.
Section 1.3: Operations with sets: Video. Notes.
Section 1.4: Functions: Video. Notes.
Section 1.6: Cardinality: Video. Notes.
Chapter 2
Section 2.2: Min, max, inf, and sup: Video. Notes.
Section 2.2: liminf and limsup: Video. Notes.
Section 2.2: Completeness of \(\Bbb{R}\) and \(\Bbb{R}^n\): Video. Notes.
Section 2.3: Four theorems from calculus: Video. Notes.
Chapter 3
Section 3.1: Metric spaces: Video. Notes.
Section 3.2: Convergence: Video. Notes.
Section 3.2: Continuity: Video. Notes.
Section 3.3: Open and closed sets: Video. Notes.
Section 3.3: Continuity in terms of open and closed sets: Video. Notes.
Section 3.4: Completeness: Video. Notes.
Section 3.4: Banach's fixed point theorem: Video. Notes.
Section 3.5: Compactness I: Video. Notes.
Section 3.5: Compactness II: Video. Notes.
Section 3.5: Compactness III: Video. Notes.
Section 3.6: Alternative description of compactness: Video. Notes.
Section 3.7: Completions: Video. Notes.
Chapter 4
Section 4.1: Modes of continuity: Video. Notes.
Section 4.2: Modes of convergence: Video. Notes.
Section 4.3: Integrating sequences of functions: Videos. Notes.
Section 4.3: Differentiasting sequences of functions: Videos. Notes.
Section 4.4: Power series: Videos. Notes.
Section 4.5: Spaces of bounded function: Videos. Notes.
Section 4.6: Spaces of continuous functions I: Videos. Notes.
Section 4.7: Ordinary differential equations I: Video. Notes.
Section 4.7: Ordinary differential equations II (uniqueness): Video. Notes.
Section 4.7: Ordinary differential equations III (existence): Video. Notes.
Section 4.8: Arzela-Ascoli's Theorem: Video. Notes.
Section 4.9: Convergence of Euler's method: Video. Notes.
Section 4.10: Weierstrass' approximation theorem I: Video. Notes.
Section 4.10: Weierstrass' approximation theorem II: Video. Notes.
Chapter 5
Chapter 5 background video: Normed spaces: Video. Notes.
Chapter 5 background video: Continuity and convergence in normed spaces: Video. Notes.
Section 5.1: Normed vector spaces I (definitions): Video. Notes.
Section 5.1: Normed vector spaces II (equivalent norms): Video. Notes.
Section 5.2: Series and bases: Video. Notes.
Section 5.3: Inner product spaces: Video. Notes.
Section 5.4: Linear operators I. Video. Notes.
Section 5.4: Linear operators II (boundedness): Video. Notes.
Section 5.4: Linear operators III (spaces of linear operators): Video. Notes.
Section 5.5: Invertible linear operators I: Video. Notes.
Section 5.5: Invertible linear operators II (Neumann series): Video. Notes.
Chapter 6
Chapter 6 background video: What is a derivative: Video. Notes.
Section 6.1: The Frechet derivative I: Video. Notes.
Section 6.1: The Frechet derivative II: Video. Notes.
Section 6.2: The Gateaux derivative: Video. Notes.
Section 6.3: The mean value theorem: Video. Notes.
Section 6.5: Multiindices: Video. Notes.
Section 6.5: Taylor's formula (in \(\Bbb{R}^d\)): Video. Notes.
Section 6.6: Partial derivatives: Video. Notes.
Section 6.7: The inverse function theorem: Video. Notes.
Section 6.8: The implicit function theorem I (motivation): Video. Notes.
Section 6.8: The implicit function theorem II (proof): Video. Notes.
Chapter 10
Section 10.1: Fourier series (introduction and motivation): Video. Notes.
Section 10.2: Convergence of Fourier series I: Video. Notes.
Section 10.2: Convergence of Fourier series II: Video. Notes.