Lectures related to Monotone convergence theorem

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

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

Applications of Convergence Theorems

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

Pointwise Convergence of Fourier Series

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

Weak Convergence in Hilbert Spaces

Explores weak convergence in Hilbert spaces, discussing definitions, implications, and examples.

Properties of Complete Spaces

Covers the properties of complete spaces, including completeness, expectations, embeddings, subsets, norms, Holder's inequality, and uniform integrability.

Subsequences and Bolzano-Weierstrass Theorem

Covers the proof of the Squeeze Theorem, Quotient Criteria, and the Bolzano-Weierstrass Theorem.

Semicontinuous Functions

Covers the concept of semicontinuous functions and their integration.

Calculus of Variations: Principles and Applications

Explores the principles and applications of calculus of variations, focusing on uniform boundedness and equi-integrability of Carathéodory integrands.

Comparison Theorem: Convergence of Sequences

Explains the comparison theorem for sequence convergence with examples and proofs.

Gibbs measures: introduction

Introduces Gibbs measures to describe systems on an infinite lattice.

Subsequences and Bolzano-Weierstrass Theorem

Covers the squeeze theorem, monotone sequences, subsequences, and Bolzano-Weierstrass theorem, emphasizing the importance of peaks in sequences.

Convergence Criteria

Covers convergence criteria for sequences and explores monotone sequences and the two policemen criterion.

Lebesgue Integral: Properties and Convergence

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

Advanced Analysis I: Monotone Bounded Sequences

Covers the concept of monotone and bounded sequences, discussing their convergence and majorants.

Quadratic Penalty Method: Finer Analysis

Covers the quadratic penalty method and augmented Lagrangian, including the setup and convergence of sequences.

Distribution Interpolation Spaces

Covers the proof of UE Lipschitz constant and distribution interpolation spaces.

The Banach Fixed Point Theorem

Explores the Banach Fixed Point Theorem, showing the uniqueness of fixed points in contraction mappings.

Dirichlet Series

Explores Dirichlet series convergence properties and absolute convergence conditions, with examples and applications.

Sequence Uniformity: Convergence and Squeeze Theorem

Explores sequence uniformity, convergence, and the Squeeze Theorem in mathematical analysis.