Lectures related to Well-Posedness of Elliptic PDEs

Weak Formulation of Elliptic PDEs

Covers the weak formulation of elliptic partial differential equations and the uniqueness of solutions in Hilbert space.

Functional Analysis I: Norms and Bounded Operators

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

Functional Analysis I: Operator Definitions

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

Linear Operators: Boundedness and Spaces

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

Continuity and Galerkin Method

Introduces continuity in function spaces and the Galerkin method for solving boundary value problems.

Bounded Operators: Theory and Applications

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

Linear Operators: Basis Transformation and Eigenvalues

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

Dual Spaces: Definitions and Properties

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

Quantum Mechanics: Power Series Solutions

Explores power series solutions in quantum mechanics for differential equations and energy quantization.

Linear Operators: Traces and Boundedness

Covers the concept of linear operators, focusing on traces and boundedness.

Postulates of Quantum Mechanics

Explains the postulates of Quantum Mechanics, focusing on self-adjoint operators and mathematical notation.

Functional Analysis I: Spectral Theorem

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

Linear Operators: Quantum Mechanics and Linear Algebra

Explores the role of linear operators in Quantum Mechanics and linear algebra, emphasizing eigenvalues and basis transformations.

Distributional Derivatives

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

Semigroups of Linear Operators

Explores semigroups of linear operators, contraction semigroups, infinitesimal generator, and Riemann integral properties.

Linear Operators: Boundedness and Convergence

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

Resolving Operators: Closedness and Injectivity

Discusses resolving operators' closedness and injectivity in Banach spaces.

Finite Element Method: Weak Formulation and Galerkin Method

Explores the weak formulation and Galerkin method in Finite Element Method applications, including boundary conditions and linear systems of equations.

Matrix Representation of Operators and Basis Transformation

Explores the matrix representation of operators and basis transformation in linear algebra.

Adjoint of Linear Operators on Inner Product Spaces

Explores the adjoint of linear operators on inner product spaces, including self-adjoint, unitary, and normal operators.