Functional Analysis

This course provides the mathematical foundations needed for the analysis of infinite-dimensional systems, with applications to operator theory, quantum mechanics, and quantum information. The course develops key tools of functional analysis—duality, weak topologies, compact operators, and fundamental theorems on bounded linear maps—forming the basis for advanced topics such as C*-algebras, von Neumann algebras, quantum states, and completely positive maps.

1. Knowledge

The student has deep knowledge of central results and concepts in functional analysis, including:

• The Hahn-Banach theorem
• The Open Mapping and Closed Graph theorems
• The Banach-Steinhaus theorem (Uniform Boundedness Principle)
• Dual spaces and weak/weak* convergence
• The Banach-Alaoglu theorem and compactness in dual spaces
• The spectral theorem for compact self-adjoint operators

The student understands how these results form the basic framework for operator algebras and the mathematical foundations of quantum theory and quantum information.

2. Skills

The student is able to:

• apply fundamental theorems of functional analysis to solve mathematical problems involving linear operators on infinite-dimensional spaces,
• use duality, weak convergence and compact operator theory in the study of operator algebras and quantum states,
• analyze the structure of bounded linear operators as they arise in quantum mechanics and quantum information theory (e.g., density operators, measurements, compact approximations, and spectral decompositions).

Lectures, exercises and a written final examination.