Lecture 1: Introductions and the definition of a metric space.
Lecture 2: The Euclidean metric and the Cauchy-Schwartz inequality. Sequences and convergence.
Lecture 3: Introduction to the concepts of topology: open and closed sets.
Lecture 4: Completeness and introduction to compactness
Lecture 5: Compactness, part 1 (the Lebesgue covering lemma)
Lecture 6: Compactness, part 2 (compactness is sequential compactness)
Lecture 7: Continuity of maps between metric spaces
Lecture 8: Continuity and compactness
Lecture 9: Extending continuous maps via uniform continuity
Lecture 10: Towards differentiation (linear algebra review)
Lecture 11: Towards differentiation (normed vector spaces)
Lecture 12: Operator norm and the definition of a derivative
Lecture 13: Derivatives and partial derivatives
Lecture 14: Computing some derivatives
Lecture 15: More derivative computation, sketch of inverse function theorem
Lecture 16: Mean values and bits of the inverse function theorem
Lecture 17: The inverse function theorem and a summary of the implicit function theorem
Lecture 19: Towards spaces of functions
Lecture 20: Uniform convergence
Lecture 21: Exchange of limits
Lecture 22: Summary of ideas
