Functional Analysis I

Course Description

 The main idea of functional analysis is that some problems in analysis can be solved by studying linear maps between vector spaces of functions. Such spaces are, in the vast majority of the cases, infinite dimensional, which means that linear functions between them can not be understood via finite matrices. Nonetheless, by endowing a vector space  with a norm (i.e., an abstract notion of length of its elements), one has a metric (and hence a topology) to work with, and tools from analysis of metric spaces and topology generally apply. For instance, one can talk about continuity of linear maps between infinite dimensional vector spaces, and can define a notion of convergence of a sequence in a vector space. Therefore, throughout this course we will use ideas from linear algebra, analysis, and topology. 

The topic that we will cover include some of the landmark results in classic functional analysis, such as:

• Normed spaces and linear operators between them
• Dual spaces and the Hahn-Banach theorem
• The open mapping theorem, the uniform  boundedness principle and the closed graph theorem
• Weak * topologies and the Banach-Alaoglu theorem
• Hilbert spaces
• Spectral theory

Prerequisite

Linear algebra, metric spaces and a few basic notions from topology. Some knowledge from a first course of measure theory are recommended, though not required.

Language

The lectures will be given in English. The problem sessions will be given in English, unless the Assistant and all the students agree to have them in German.

Lectures

Tuesday and Thursday, 14:00 - 16:00 in HS IV, building E2.4

Problem Sessions

To be decided, two hours a week starting with week 2

Office Hours

To be decided, two hours a week

Assignments and Exam

There will be weekly homework assignments. The exam will be oral, and in order to access it one has to obtain at least the half of the available points from the homework assignments.

Contact:

For any question regarding the course, don't hesitate to contact the instructor at dayan@math.uni-sb.de