M615: Introduction to Classical Analysis

Course Materials:

Syllabus
Syllabus Video (11:53)
eCampus Video (11:12)

Preliminary Lectures:

Elements of Set Theory, I (8:19); Full lecture PDF
Elements of Set Theory, II (7:35); Full lecture PDF

Week 1 Lectures: Chapter 1, Part I

Overview and the Least Upper Bound Axiom (9:14); Full lecture PDF
Some Consequences of the Least Upper Bound Axiom (11:04); Full lecture PDF
Monotone Convergence Theorem (12:58); Full lecture PDF
The Nested Interval Theorem (9:29); Full lecture PDF
Geometric Series and Proposition 1.7 (9:25); Full lecture PDF
P-adic Expansions (16:06); Full lecture PDF
Some Fine Print about P-adic Expansions (8:51); Full lecture PDF
Bernoulli's Inequality (4:53); Full lecture PDF
Euler's Number (14:28); Full lecture PDF

Week 2 Lectures: Chapter 1, Part II

Limit Superior and Limit Inferior, I: Bounded Sequences (8:50); Full lecture PDF
Limit Superior and Limit Inferior, II: Unbounded Sequences (11:32); Full lecture PDF
Bolzano-Weierstrass Theorem (12:36); Full lecture PDF
Cauchy Sequences (11:38); Full lecture PDF
Limits of Functions (7:58); Full lecture PDF
Continuity (8:55); Full lecture PDF
Monotone Functions (11:21); Full lecture PDF

Week 3 Lectures: Chapter 2, Part I

Set Equivalence (9:12); Full lecture PDF
Equivalence of \mathbb{N} \times \mathbb{N} and \mathbb{N} (6:42); Full lecture PDF
Infinite Subsets of \mathbb{N} are Countable (9:50); Full lecture PDF
Sequences in \mathbb{R} have Monotone Subsequences (11:02); Full lecture PDF
Countable Unions of Countable Sets (13:37); Full lecture PDF
Uncountable Sets (8:26); Full lecture PDF
Power Sets (7:45); Full lecture PDF
Cantor's Theorem (11:07); Full lecture PDF
F. Bernstein's Theorem (20:10); Full lecture PDF
Cardinality of P(\mathbb{N}) (18:09); Full lecture PDF

Week 4 Lectures: Chapter 2, Part II

The Cantor Set (11:36); Full lecture PDF
Cardinality of the Cantor Set (15:21); Full lecture PDF
P-adic Characterization of the Cantor Set (15:39); Full lecture PDF
The Cantor Function (14:12); Full lecture PDF
A Second Proof that \rm{card} (\Delta) = c (10:22); Full lecture PDF
Extension of the Cantor Function (13:42); Full lecture PDF
Generalized Cantor Sets and \Delta - \Delta (10:00); Full lecture PDF
\Delta - \Delta = [-1,1] (16:26); Full lecture PDF
Some Properties of Monotonic Functions (13:52); Full lecture PDF
The Extended Cantor Function is Continuous (14:53); Full lecture PDF

Week 5 Lectures: Chapter 3

Familiar Metrics and Norms (10:49); Full lecture PDF
Metric Spaces (12:32); Full lecture PDF
Normed Vector Spaces (13:39); Full lecture PDF
The Cauchy-Schwarz Inequality for \ell_2 (17:33); Full lecture PDF
The Triangle Inequality for \ell_2 (6:58); Full lecture PDF
Norms on \ell_p Spaces, I (21:37); Full lecture PDF
Norms on \ell_p Spaces, II (23:36); Full lecture PDF
Limits in Metric Spaces (15:16); Full lecture PDF
Cauchy Sequences for Metric Spaces (9:23); Full lecture PDF
\mathbb{R}^n with the Euclidean Metric, and \ell_2 (12:05); Full lecture PDF

Week 6 Lectures: Chapter 4

Open Sets (12:12); Full lecture PDF
Unions and Intersections of Open Sets (12:05); Full lecture PDF
Open Subsets of \mathbb{R} (13:11); Full lecture PDF
Sequential Characterization of Open Sets (8:46); Full lecture PDF
Closed Sets (12:06); Full lecture PDF
Sequential Characterization of Closed Sets (12:42); Full lecture PDF
Interior and Closure (9:14); Full lecture PDF
Sequential Characterization of Closure (9:32); Full lecture PDF
The Relative Metric (14:18); Full lecture PDF
Open and Closed Sets in the Relative Metric (9:48); Full lecture PDF

Week 7 Lectures: Chapter 5

Continuity in Metric Spaces (14:13); Full lecture PDF
Characterizations of Continuity (15:26); Full lecture PDF
Distance from a Point to a Set (16:38); Full lecture PDF
Homeomorphisms (16:18); Full lecture PDF
Examples of homeomorphisms (17:34); Full lecture PDF
Note on Topology (8:39); Full lecture PDF
The Space of Continuous Functions (14:36); Full lecture PDF
Algebras, Groups, Rings, and Lattices (15:46); Full lecture PDF

Week 8 Lectures: Chapter 6

Connectedness (12:17); Full lecture PDF
Precise Definition of Connectedness (7:43); Full lecture PDF
Disconnections (13:56); Full lecture PDF
Proof of Lemma 6.3 (9:13); Full lecture PDF
Connected Subsets of \mathbb{R} (15:55); Full lecture PDF
Connectedness and Continuous Functions (10:48); Full lecture PDF
The Intermediate Value Theorem (11:44); Full lecture PDF
If A and B are Connected, A \times B is Connected (10:53); Full lecture PDF
Is \mathbb{R} Homeomorphic to \mathbb{R}^2 ? (12:38); Full lecture PDF
Space-Filling Curves (12:55); Full lecture PDF
Lebesgue's Space-Filling Curve (10:44); Full lecture PDF

Week 9 Lectures: Chapter 7, Part I

Totally Bounded Sets (10:36); Full lecture PDF
Alternative Formulation of Total Boundedness (13:06); Full lecture PDF
Cauchy Sequences and Total Boundedness, I (20:38); Full lecture PDF
Cauchy Sequences and Total Boundedness (12:52); Full lecture PDF
Complete Metric Spaces (8:13); Full lecture PDF
\ell_2 is Complete (14:08); Full lecture PDF
Subsets of Complete Metric Spaces (4:44); Full lecture PDF
Characterizations of Complete Metric Spaces, I (9:48); Full lecture PDF
Characterizations of Complete Metric Spaces, II (12:32); Full lecture PDF

Week 10 Lectures: Chapter 7, Part II

Banach Spaces (8:43); Full lecture PDF
Series Characterization of Banach Spaces (17:03); Full lecture PDF
Newton's Method (16:00); Full lecture PDF
Matrix Equations (11:20); Full lecture PDF
Ordinary Differential Equations (6:37); Full lecture PDF
Contraction Mapping Theorem (14:04); Full lecture PDF
Proof of the Contraction Mapping Theorem (17:55); Full lecture PDF
Contraction Maps on \mathbb{R} and \mathbb{R}^n (14:52); Full lecture PDF
Application to Newton's Method (13:14); Full lecture PDF
Application to Linear Systems (11:31); Full lecture PDF

Week 11 Lectures: Chapter 7, Part III

Application to ODE's, I (12:14); Full lecture PDF
Application to ODE's, II (20:20); Full lecture PDF
Completions (9:43); Full lecture PDF
The Space \ell_{\infty} (M) (10:22); Full lecture PDF
M is Isometric to a Subset of \ell_{\infty} (M) (16:28); Full lecture PDF
The Completion of (M,d) (8:12); Full lecture PDF
Uniqueness of Completions (25:16); Full lecture PDF
Completions of Normed Vector Spaces (13:09); Full lecture PDF

Week 12 Lectures: Chapter 8, Part I

Compactness (7:49); Full lecture PDF
Sequential Characterization of Compactness (11:03); Full lecture PDF
Continuous Functions Map Compact Sets to Compact Sets (17:09); Full lecture PDF
Topological Characterization of Compactness, I (14:41); Full lecture PDF
Topological Characterization of Compactness, II (18:41); Full lecture PDF
Two More Characterizations of Compactness (13:06); Full lecture PDF
Uniform Continuity (7:50); Full lecture PDF
Alternative Characterizations of Uniform Continuity (10:25); Full lecture PDF
Continuous Functions on Compact Metric Spaces (10:00); Full lecture PDF

Week 13 Lectures: Chapter 8, Part II

Extension of Uniformly Continuous Functions (25:47); Full lecture PDF
Uniqueness of Completions (a Shorter Proof) (9:03); Full lecture PDF
Equivalent Metrics (17:29); Full lecture PDF
Equivalent Metrics on Compact Metric Spaces (5:22); Full lecture PDF
Properties of Continuous Linear Maps (11:39); Full lecture PDF
Two Space of Bounded Linear Maps (18:35); Full lecture PDF
Equivalence of Norms on a Single Vector Space (7:29); Full lecture PDF
Equivalence of Norms on Finite-Dimensional Vector Spaces (28:56); Full lecture PDF
Properties of Finite-Dimensional Vector Spaces (14:39); Full lecture PDF

Week 14 Lectures: Chapter 9

The Oscillation of a Function on an Interval (18:03); Full lecture PDF
The Oscillation of a Function at a Point (13:10); Full lecture PDF
A Characterization of D(f) (7:50); Full lecture PDF
The Baire Category Theorem for \mathbb{R} (12:44); Full lecture PDF
Applications of the BCT on \mathbb{R} , I (10:44); Full lecture PDF
Applications of the BCT on \mathbb{R} , II (14:43); Full lecture PDF
Nowhere Dense Sets (14:02); Full lecture PDF
The Baire Category Theorem (20:05); Full lecture PDF
The Complement of a First Category Set is Dense (5:37); Full lecture PDF