Functional Analysis I: Consequences of Baire Theorem

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Optimal Transport: Heat Equation and Metric Spaces

Explores optimal transport in heat equations and metric spaces.

The Banach Fixed Point Theorem

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

Probability Theory: Laws and Convergence

Covers Borel-Cantelli lemmas, laws of random variables, and convergence in probability.

Optimal Transport: Stability and Counterexamples

Covers stability in optimal transport and provides examples of discontinuous optimal solutions.

Properties of Complete Spaces

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

Metric Spaces: Norms and Distances

Explores norms, distances, scalar products, and norm convergence in metric spaces.

Normed Spaces & Reflexivity

Covers normed spaces, Banach spaces, and Hilbert spaces, as well as dual spaces and weak convergence.

Cauchy-Lipschitz Theorem: Examples and Applications

Explores examples and applications of the Cauchy-Lipschitz theorem in sequences and Banach spaces.

Functional Analysis: Banach and Hilbert Spaces

Covers Banach and Hilbert spaces, separability, norm, continuity, and functional analysis.

Convergence of Cauchy Sequences: Theorem Applications

Covers the convergence of Cauchy sequences and its applications in theorems.