Lectures related to Analysis IV: Convergence and Approximation in L² Space

Discusses approximation by smooth functions and the convergence of function sequences in normed vector spaces.

Covers the preliminaries in measure theory, including loc comp, separable, complete metric space, and tightness concepts.

Continuous Functions: Criteria and Equivalences

Explores criteria for continuity, convergence of sequences, and equivalence conditions for continuous functions.

Function Approximations

Covers continuous functions with compact support, density, and approximation, focusing on the heat equation.

Bolzano-Bastra-Sterl: Applications and Uniform Continuity

Explores the Bolzano-Bastra-Sterl theorem and uniform continuity in sequences.

Advanced Analysis II: March 23

Covers variable bounds, convergence of integrals, and homomorphisms in advanced analysis.

Generalized Integrals: Type 2

Covers the integration of limit expansions and continuous functions by pieces.

Analysis: Recap and Normed Space R^n

Covers a recap of Analysis 1 and 2, emphasizing normed space R^n, subsets, and continuous functions.

Global Properties of Continuous Functions

Explores the global properties of continuous functions, including behavior, limits, and interval characteristics.

Convergence and Closed Sets

Explores convergence of sequences in closed sets and the importance of understanding convergence in relation to closedness.

Fourier Series: Convergence and Dirichlet Theorem

Covers Fourier series convergence, Dirichlet theorem, and applications in signal processing.

Generalized Integrals: Bounded Interval

Introduces generalized integrals on a bounded interval, discussing convergence, divergence, comparison criteria, variable substitution, and corollaries.

Advanced Analysis I: Continuous Functions on Compact Sets

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

Normed Spaces

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

Multiple Integration: Part 2

Explores double integrals over unbounded domains and convergence of sequences.

Limit of Functions: Convergence and Boundedness

Explores limits, convergence, and boundedness of functions and sequences.

Riesz-Fischer Theorem

Explores the Riesz-Fischer theorem, discussing completeness and convergence in L^p spaces with examples and demonstrations.

Interpolation of a Continuous Function

Explores the interpolation of a continuous function by a polynomial.

Weak Derivatives: Definition and Properties

Covers weak derivatives, their properties, and applications in functional analysis.

Advanced Analysis I: Uniform Convergence

Covers the concept of uniform convergence of continuous functions towards a limit function.