Lectures related to Advanced analysis II: riemann integral properties

Explores Fubini's Theorem for multiple integrals, emphasizing the n=2 case.

Compact Embedding: Theorem and Sobolev Inequalities

Covers the concept of compact embedding in Banach spaces and Sobolev inequalities.

Covers multivariable integral calculus, including rectangular cuboids, subdivisions, Douboux sums, Fubini's Theorem, and integration over bounded sets.

Fubini Theorem on Closed Rectangles

Explores the Fubini theorem on closed rectangles in R², discussing integrability, iterated integrals, and compact sets.

Interior Points and Compact Sets

Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

Compact Sets and Extreme Values

Explores compact sets, extreme values, and function theorems on bounded sets.

Advanced Analysis II: Riemann Integrability and Jordan Measure

Explores Riemann integrability and Jordan measure, discussing the conditions for a set to be negligible.

Normed Spaces

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

Integral Properties on Closed Pavés

Explores the integrability of continuous functions on closed pavés and the properties of their integrals, including boundedness and Darboux sums.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Riemann Integral: Construction and Properties

Explores the construction and properties of the Riemann integral, including integral properties and mean value theorem.

Convergent Sequences: Definitions and Illustrations

Explains convergent sequences, bounded sequences, subsequences, and compact sets with illustrations and proofs.

Maximum and Minimum of Functions

Explores limits, minimum and maximum values of functions, and continuity criteria in a compact set.

Functional Analysis I: Operator Definitions

Introduces linear and bounded operators, compact operators, and the Banach space.

Taylor Series and Riemann Integral

Explores Taylor series expansions and Riemann integrals, including limits, convergence, subdivisions, and sums.

Multiple Integrals: Definitions and Properties

Covers the definition and properties of multiple integrals, including double and triple integrals.

Convergence and Limits in Real Numbers

Explains convergence, limits, bounded sequences, and the Bolzano-Weierstrass theorem in real numbers.

Mild Dissipative Surface Dynamics

Explores mild dissipative surface dynamics, including conservative behavior and ergodic functions.

Advanced Analysis II: Jordan-Measurable Functions

Explores Jordan-measurable functions and double integrals for volume calculations in 3D space.

Compact Subsets of R^n

Explores compact subsets of R^n, convergence theorems, and set properties.