Lectures related to Dominated convergence theorem

Explores monotone convergence, dominated convergence, and Fatou's lemma with practical examples.

Explores applications of dominated convergence and the Fatou lemma in real analysis.

Explores uniform integrability, convergence theorems, and the importance of bounded sequences in understanding the convergence of random variables.

Pointwise Convergence of Fourier Series

Explores the pointwise convergence of Fourier series and its applications in optimal transport.

Lebesgue Integral: Definition and Properties

Explores the Lebesgue integral, where functions self-select partitions, leading to measurable sets and non-measurable complexities.

Fourier Transform and Schwartz Space

Explores the Fourier transform, Schwartz space, and Riemann-Lebesgue lemma in quantum physics.

Differentiation under Integral Sign

Explores differentiation under the integral sign, comparing it with the Riemann integral and discussing key assumptions and theorems.

Finite Elements Method: Error Estimation

Explores a priori error estimation in the finite elements method, covering convergence analysis, orthogonality, weak formulations, and optimal precision.

Riemann Integral: Convergence and Limit Process

Explores Riemann integral, convergence, and limit processes, emphasizing continuity and monotonic convergence.

Advanced Analysis I: Uniform Convergence Theorem

Covers the Uniform Convergence Theorem and its applications to integrals and function spaces.

Approximation by Smooth Functions

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

Convergence of Integrals: Criteria and Examples

Explores the convergence of integrals through criteria and examples, emphasizing the importance of understanding both sides' convergence.

Nonlinear Equations: Fixed Point Method Convergence

Covers the convergence of fixed point methods for nonlinear equations, including global and local convergence theorems and the order of convergence.

Harmonic Series Divergence

Covers harmonic series divergence and geometric series convergence and divergence with demonstrations.

Exemple: Convergence Analysis

Analyzes the convergence of a function using the Newton method.

Lebesgue Integral: Properties and Convergence

Covers the Lebesgue integral, properties, and convergence of functions.

Convergence in Law: Theorem and Proof

Explores convergence in law for random variables, including Kolmogorov's theorem and proofs based on probability lemmas.

Lebesgue Measure and Fourier Analysis

Explores Lebesgue measure, Fourier analysis, PDE applications, and optimal transport in PDEs.

Martingale Convergence Theorems

Explores the convergence of martingales under specific conditions and previews upcoming topics on martingale theorems and inequalities.

Iterative Methods: Linear Systems

Covers iterative methods for solving linear systems, emphasizing convergence analysis and well-chosen matrices.