Lectures related to Stone-Weierstrass Theorem

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

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

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

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

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

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

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

Covers the Hadamard factorisation theorem for entire functions of order at most 1.

Explores the necessity of uniform continuity for continuous functions on compact sets.

Introduces Laurent series in complex analysis, focusing on convergence and analytic functions.

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Explores compact sets, extreme values, and function theorems on bounded sets.

Explores norm equivalence in complex functions, covering homogeneity and triangular inequality.

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

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

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

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

Discusses holomorphic functions, focusing on the Cauchy-Riemann equations and their applications in complex analysis.

Discusses approximation by smooth functions and the convergence of function sequences in normed vector spaces.