Concept

Uniform space

Related lectures (31)

Differential Equations: Solutions and Periodicity

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

Covers classical testing of qubits, including measuring in different bases and obtaining uniform results.

Explores limits, continuity, and uniform continuity in functions, including properties at specific points and closed intervals.

Intermediate Value Theorem

Covers the Intermediate Value Theorem, uniform continuity, Lipschitz functions, and the properties of continuous functions.

Magnetic Induction

Explains magnetic induction, force on a dipole, torque, and applications in macroscopic systems.

Untitled

Advanced Analysis I: Continuous Functions on Compact Sets

Explores the necessity of uniform continuity for continuous functions on compact sets.

Uniform Continuity: Definitions and Examples

Explores uniform continuity in functions, covering definitions, examples, and properties.

Multiple Integration: Fubini Theorem

Explores multiple integration in R², focusing on double integrals over closed rectangles and the Fubini theorem.

Riemannian distance, geodesically convex sets

Covers the structure of Riemannian manifolds, geodesic convexity, and the Riemannian distance function.

Hyperbolic Geometry

Introduces hyperbolic geometry, covering complete metric spaces, isometries, and Gaussian curvature in dimension 2.

Probability Theory: Laws and Convergence

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

Cauchy Test and Generators

Explores Cauchy test, generators, and density in functional analysis.

Integrals on Unbounded Domains

Explores integrals on unbounded domains, addressing challenges with discontinuities and providing examples for illustration.

Definition of Definite Integral

Explains the definition of the definite integral and the independence of subdivisions.

Ordinary Differential Equations

Covers the basics of ordinary differential equations and their properties.

Topology: Functions and Solutions

Explores topology, functions, and solutions, emphasizing reduction to Cauchy and uniformization.

Properties of Complete Spaces

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

Real Functions: Continuity and Limits

Explores continuity and limits of real functions, including examples and the concept of uniform continuity.

Initial Problem Solutions

Covers the description of problem solutions and the concept of compactness and uniform continuity.