Lectures related to Theorems in Analysis

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

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

Explores distribution and interpolation spaces, showcasing their importance in mathematical analysis and the computations involved.

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

Explains the Wronskian and its role in determining linear independence of solutions to differential equations.

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

Covers the approximation of functions in Sobolev spaces using smooth functions.

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Explains the Wronskian and its role in determining linear independence of solutions to differential equations.

Explores differentiating under the integral sign and continuity of functions in integrals.

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

Explores optimal transport analysis and proofs, emphasizing weak convergence and compactness.

Covers the proof of the existence of y and the resolution of EDOs with practical examples.

Explores Sobolev spaces in higher dimensions, discussing derivatives, properties, and challenges with continuity.

Covers the Implicit Functions Theorem, providing a general understanding of implicit functions.

Explores sub/super harmonic functions and their applications in a theoretical context.

Covers the properties of harmonic functions and the concept of mollification.

Introduces logic, proofs, sets, functions, and algorithms in mathematics and computer science.

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

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.