Super Secret Functional Analysis

Functional analysis is a branch of mathematics concerned with infinite-dimensional vector spaces (mainly function spaces) and mappings between them. The spaces may be of different, and possibly infinite, dimensions. These mappings are called operators or, if the range is on the real line or in the complex plane, functionals.
It employs mathematical tools from analysis, topology, measure theory, linear algebra, and abstract algebra.
The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. Applications include Itô stochastic calculus, Economic Theory, Quantum Mechanics and many others.

### Course Overview

This course in functional analysis covers topics as one would encounter in an (under)graduate course in functional analysis.

Course topics include normed spaces, linear functionals, linear operators, inner product spaces(Hilbert spaces), strong and weak convergence, Banach fixed point theorem, Chebyshev polynomials, spectral theory, etc.

### Format and Scope

Lectures will be offered on an on-demand basis(?). Math discussion will be facilitated on the private Stack Exchange (for formal, on-topic discussion) and on the Element server (for informal on-topic discussion). Collaborative notes and on-topic information sharing (examples, explanations for concepts, etc.) will be facilitated here on the Fandom.

The scope of our study in functional analysis with respect to the contents of the book is partitioned into X main phases:

#### Chapter 1-3 - Spaces and Operators

* Metric Spaces
* Normed and Banach spaces
* Linear operators
* Inner product and Hilbert spaces

#### Chapter 4 - Fundamental Theorems

* Hahn-Banach theorem
* Uniform boundedness threoem
* Open mapping theorem
* Closed graph theorem

#### Chapter 5-6(optional) - Further Applications

* Applications of contractions
* Approximation theory

#### Chapter 7-9 - Spectral Theory

* Basic concepts
* Operators on normed spaces
* Compact operators
* Self-adjoint operators

#### Chapter 10-11 - Unbounded operators

* Unbounded operators
* Quantum mechanics

### Textbook

We will be using the textbook *Introductiory Functional Analaysis with Applications* by Erwin Kreyszig([1]).

### Coursework

Those who take part in studying functional analysis are highly encouraged to solve problems. Homework will be "assigned" each week, along with a reading "assignment". At the end of each chapter... and stuff may be assigned...

### Prerequisites and Difficulty

A background in undergraduate mathematics, in particular, linear algebra and ordinary calculus, is sufficient as a prerequisite. Measure theory is neither assumed nor discussed. No knowledge in topology is required; the few considerations involving compactness are self-contained. Complex analysis is not needed, except in one of the later sections (Sec. 7.5), which is optional, so that it can easily be omitted. Further help is given in Appendix 1, which contains simple material for review and reference. The book should therefore be accessible to a wide spectrum of students and may also facilitate the transition between linear algebra and advanced functional analysis.