Advanced Analysis

1. Introduction to Analysis

Unit 1: The Real Numbers (2+2 Sessions)
(Video) Complete Ordered Fields
01 - Completeness and the Supremum
(Video) The Natural Numbers
02 - Counting and Cardinality
Unit 2: Sequences and Series (2+4 Sessions)
(Video) Sequences Introduction
03 - Sequence Properties and Examples
04 - Monotone Sequences
05 - Subsequences and Bolzano-Weierstrass
06 - Part 1 - Cauchy Sequences
(Video) Series Introduction
06 - Part 2 - Infinite Series
(Supplement) Conditional Convergence and the Riemann Rearrangement Theorem
Unit 3: Topology of the Reals (1+2 Sessions)
(Video) Topology
07 - Working with Topology in \({\mathbb R}\)
08 - Compact and Connected Sets
Unit 4: Functional Limits (1+3 Sessions)
(Video) Overview of Limits, Continuity, and Uniform Continuity
09 - Limit Laws, Sequential Characterization, and Uniform Continuity
10 - The Extreme Value Theorem
11 - The Intermediate Value Theorem
Unit 5: Derivatives (2+2 Sessions)
12 - Derivatives: Definitions and Fundamental Results
13 - Rolle's Theorem, Mean Value Theorem, and Generalized MVT
(Video) Inverse Functions
(Video) Convexity
Unit 6: Sequences and Series of Functions (2+4 Sessions)
(Video) Uniform Convergence
14 - Uniform Convergence and Differentiability
15 - Power Series and Taylor Series
16 - Rigorous Development of Trigonometric Functions
(Video) Arzelà-Ascoli Theorem
17 - Weierstrass Approximation Theorem
Unit 7: Riemann Integration (2+2 Sessions)
(Video) Integration, Part I
18 - Properties of the Integral; Products; Uniform Limits
(Video) Integrable Functions
19 - The Fundamental Theorem of Calculus; Substitution
(Supplement) Riemann versus Darboux Integration
(Supplement) Differentiating Integrals
Unit 8A: Metric Spaces (1+4 Sessions)
20 - Metric Spaces: Definitions and Illustrative Examples
21 - Continuity and Compactness; Completeness Defined
22 - Completeness and Compactness; Baire Category Theorem
(Video) Metric Space Completions
23 - Contraction Mapping Principle
Unit 8B: Fourier Series (1+3 Sessions)
(Video) Fourier Series Introduction
24 - Convergence of Partial Sums
25 - Summability Methods
26 - Mean Square Convergence and Parseval's Identity

2. Further Topics in Analysis

Unit 9: Applications of Fourier Series (1+2 Sessions)
(Video) The Isoperimetric Inequality
01 - Weyl's Criterion for Equidistribution
02 - Continuous, Nowhere Differentiable Functions
Unit 10: Hilbert Spaces (1+2 Sessions)
(Video) Hilbert Space Definition and Basic Properties
03 - Orthogonal Complements and Projections
04 - Orthonormal Bases
Unit 11: Metric and Euclidean Spaces (2+4 Sessions)
(Video) Fundamental Inequalities
05 - Compactness and Heine-Borel
(Video) Bolzano-Weierstrass
06 - Limits and Continuity
07 - Intermediate and Extreme Value Theorems
08 - The Separating Hyperplane Theorem
(Supplement) Convex Functions on \({\mathbb R}^d\)
Unit 12: Differentiation in Euclidean Space (2+6 Sessions)
(Video) Partial Differentiation
09 - The Total Derivative
10 - The Chain Rule
(Video) Mean Value Theorems
11 - Taylor's Theorem
12 - The Inverse Function Theorem
13 - The Implicit Function Theorem
14 - Multivariate Optimization
Unit 13: Integration in Euclidean Space (2 + 6 Sessions)
(Video) Partitions and Grids
(Supplement) Further Useful Partition Results
15 - Jordan Regions I
16 - Jordan Regions II
17 - Jordan Regions III
(Video) Riemann Integration
(Supplement) Integrating Vector-Valued Functions; Jensen's Inequality
18 - Iterated Integrals
19 - Change of Variables
20 - Polar Coordinates
Unit 14: Fourier Theory (2 + 4 Sessions)
(Video) Schwartz Space
21 - The Fourier Transform
22 - Fourier Inversion
23 - The Plancherel Formula
(Video) Poisson Summation
24 - Discrete Fourier Analysis
Unit 15: Dirichlet's Theorem (2 + 3 Sessions)
(Video) Introduction and Zeta Function
25 - Dirichlet Characters
26 - Setup of Dirichlet's Theorem
(Video) Logarithms
27 - Conclusion of the Proof of Dirichlet's Theorem