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.

We will see 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

These results form the basic framework for operator algebras and the mathematical foundations of quantum theory and quantum information.

Lecturer: Eduard Ortega

Lectures

• Thursday 12:15-14:00 in Simastuen 656 (6th floor in SBII)
• Fridays 12:15-14:00 in Simastuen 656 (6th floor in SBII)
• First Lecture Thursday 8th January

Office hours

• By appointment.

Reading materials

A. Bowers and N.J. Kalton An Introductory Course in Functional Analysis

Also see Harald Hanche-Olsen's notes and, E. Lieb and M. Loss. Analysis. Graduate Studies, AMS, for a more elementary approach to L^p-spaces, duality of L^p-spaces and related material on Lebesgues spaces that we are going to discuss during the first weeks.

Exam