Lectures related to Functional Analysis I: Operator Definitions

Normed Spaces

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

Functional Analysis I: Norms and Bounded Operators

Explores norms and bounded operators in functional analysis, demonstrating their properties and applications.

Banach Spaces: Reflexivity and Convergence

Explores Banach spaces, emphasizing reflexivity and sequence convergence in a rigorous mathematical framework.

Compact Embedding: Theorem and Sobolev Inequalities

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

Linear Operators: Boundedness and Convergence

Explores linear operators, boundedness, and convergence in Banach spaces, focusing on Cauchy sequences and operator identification.

Bounded Operators: Theory and Applications

Covers bounded operators between normed vector spaces, emphasizing the importance of continuity and exploring applications like the Fourier transform.

Functional Analysis I: Spectral Theorem

Covers the spectral theorem, orthanormal sequences, and bounded linear operators in Hilbert spaces.

Dual Spaces: Definitions and Properties

Covers the definitions and properties of dual spaces and linear operators.

The Closed Graph Theorem

Explores the Banach-Steinhaus theorem and the Closed Graph Theorem.

Linear Operators: Boundedness and Spaces

Explores linear operators, boundedness, and vector spaces with a focus on verifying bounded aspects.

Linear Operations: Convergence and Sequences

Explores linear operations, convergence, sequences, and compact sets in mathematical analysis.

Distributional Derivatives

Explores distributional derivatives, continuity, boundedness of linear operators, and weak-* continuity.

Extension of Linear Transformations

Covers the extension of bounded linear transformations and the free propagator in L^2 spaces.

Linear Operators: Basis Transformation and Eigenvalues

Explores basis transformation, eigenvalues, and linear operators in inner product spaces, emphasizing their significance in Quantum Mechanics.

Resolving Operators: Closedness and Injectivity

Discusses resolving operators' closedness and injectivity in Banach spaces.

Functional Analysis and Distribution Theory

Introduces functional analysis, distribution theory, topological vector spaces, and linear operators, emphasizing their importance in engineering applications.

Sequences and Convergence: Understanding Mathematical Foundations

Covers the concepts of sequences, convergence, and boundedness in mathematics.

Convergence and Limits in Real Numbers

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

Harmonic Forms: Main Theorem

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

Weak Derivatives: Definition and Properties

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