Lectures related to Advanced Analysis I: Continuous Functions on Compact Sets

Covers the preliminaries in measure theory, including loc comp, separable, complete metric space, and tightness concepts.

Explores dense sets, Cauchy sequences, periodic solutions, and unique solutions in differential equations.

Explores the Darboux theorem for continuous functions on closed intervals, emphasizing uniform continuity and function behavior implications.

Covers the Cantor-Heine theorem, discussing uniform continuity and compactness.

Covers the concept of uniform continuity and a theorem on continuous functions.

Covers the description of problem solutions and the concept of compactness and uniform continuity.

Delves into advanced analysis topics, emphasizing continuity, limits, and uniform continuity.

Covers continuous functions with compact support, density, and approximation, focusing on the heat equation.

Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

Explores continuous functions on compact sets, discussing boundedness and extremum values.

Explores limits, continuity, and uniform continuity in functions, including properties at specific points and closed intervals.

Explores criteria for continuity, convergence of sequences, and equivalence conditions for continuous functions.

Covers extreme values of continuous functions on compact sets and differentiability.

Explores the Stone-Weierstrass theorem, proving uniform density of specific function families on compact sets.

Covers matrix diagonalization, compact sets, continuity of functions, and the Mandelbrot set.

Covers the Intermediate Value Theorem, uniform continuity, Lipschitz functions, and the properties of continuous functions.

Covers weak derivatives, their properties, and applications in functional analysis.

Covers limits, range of continuous functions, and uniform continuity on closed intervals.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Discusses the properties of sequences, convergence, and their relationship with topology and compactness.