# Trinity College Dublin # Mathematics MA3421

## Introduction

A key goal in this module and its continuation MA3422 is to introduce at least some of the aspects of functional analysis. It is perhaps not easy to know what functional analysis is before completing a course such as this one. A (simplified) explanation is as follows.

Much of mathematics arose from the desire to solve equations (think of the square root of 2 as the length of the diagonal of a unit square, but not a rational number and so a paradox to Euclid; or complex numbers which can be said to have arisen from the need to solve general quadratic equations; diophantine equations where solutions in integers are sought and are the concern of a large part of number theory). The subject of algebraic geometry is concerned with the study of solutions of (systems of) algebraic equations, but a large part of the motivation for functional analysis is related to differential equations.

The solutions of differential equations have to be functions of some sort (sometimes continuous, more obviously differentiable functions so we can write down the differential equations). In functional analysis, we are mostly looking at the spaces that could be spaces of functions and their properties. We also look at (usually linear) operators and desirable properties they could have. A key role comes from ideas involving limits or convergence. We might like to understand when a function is "close" to a solution. For this reason we have norms on our spaces, or other mechanisms that can be used to describe convergence.

We will begin with some general topology, which is a kind of language that underlies lots of what goes on in Functional Analysis, but is also a basic thing for other subjects like Algebraic Topology or differentiable manifilds. In Functional Analysis itself, we begin with normed spaces and these are metric spaces so that one does not necessarily need general topology at the start. Some of the examples are spaces of continuous functions on topological spaces and then the more interesting examples require consideration of general topological spaces at times. There are also other topologies on (infinite dimensional) normed spaces, like the weak topology, which arise when one delves a little more deeply. So general topology is a basic tool in many ways.

See the Syllabus for more specific information.

There are past papers for this module online at the TCD Exams office archive.

Solutions to the 2014-15 paper are available here.