Functional Analysis I

Functional Analysis I

Course description

Functional analysis provides an abstract framework for the study of many questions in analysis. A basic idea behind functional analysis is to regard the objects of interest, for example sequences or functions, as points in a vector space. To obtain a meaningful theory, one endows the vector space with a suitable norm, which makes it possible to talk about concepts such as convergence or continuity. Thus, functional analysis blends together ideas from linear algebra and from analysis.

This course will provide an introduction to the fundamental concepts of functional analysis. In particular, this includes the following topics:

• Normed spaces
• Linear operators
• Dual spaces and Hahn-Banach theorem
• Open mapping theorem and uniform boundedness principle
• Hilbert spaces
• Banach and C*-algebras
• Spectral theory

Prerequisites

Basic lectures in analysis and linear algebra (e.g. Analysis 1-3, Linear Algebra 1-2), some knowledge of complex analysis.

Language

The course will be held in English unless all participants speak German.

Lecture

Monday, 14:15-15:45, in HS 3 (Building E2 5)

Thursday, 12:15-13:45, in HS 4 (Building E2 4)

Tutorial

Wednesday, 14:15-15:45, SR 10 (Building E 4)

Office Hours

Monday, 11:00-12:00, Michael Hartz, via MS Teams or in person

Tuesday, 16:00-17:00, Marcel Scherer, via MS Teams

Assignments

There will be weekly assignments.

Exam

In order to be admitted to the exam, you have to achieve at least half of the available points on the assignments. At the end of the term, there will be an exam, most likely oral.

Literature

Dirk Werner - Funktionalanalysis
John B. Conway - A Course in Functional Analysis
Peter D. Lax - Functional Analysis
Gert K. Pedersen - Analysis Now
Walter Rudin - Functional Analysis
Orr Moshe Shalit - A First Course in Functional Analysi

The library has a page for this course, where you can access many of the books electronically: Library page