Convergence and Completeness

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Related lectures (31)

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Optimal Transport: Stability and Counterexamples

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

Functional Analysis: Baire's Theorem

Covers the proof of Baire's Theorem, consequences of the Baire Category Theorem, and the principle of uniform boundedness.

Properties of Complete Spaces

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

Sequences: convergence and limits

Covers sequences, convergence, and limits in Analysis II, including Cauchy sequences and closed sets.

Differential Equations: Solutions and Periodicity

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

Completeness of LP

Explores the completeness of LP, discussing implications of mathematical conditions and the convergence of series.

The Banach Fixed Point Theorem

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

Normed Spaces

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

Cauchy-Lipschitz Theorem: Examples and Applications

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

Metric Spaces: Norms and Distances

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