II Linear Analysis - Introduction

0Introduction

II Linear Analysis

0 Introduction

In IB Linear Algebra, we studied vector spaces in general. Most of the time, we

concentrated on finite-dimensional vector spaces, since these are easy to reason

about. For example, we know that every finite-dimensional vector space (by

definition) has a basis. Using the basis, we can represent vectors and linear maps

concretely as column vectors (in F

) and matrices.

However, in real life, often we have to work with infinite-dimensional vector

spaces instead. For example, we might want to consider the vector space of

all continuous (real) functions, or the vector space of infinite sequences. It is

difficult to analyse these spaces using the tools from IB Linear Algebra, since

many of those assume the vector space is finite-dimensional. Moreover, in these

cases, we often are not interested in the vector space structure itself. It’s just

that the objects we are interested in happen to have a vector space structure.

Instead, we want to look at notions like continuity and convergence. We want to

do analysis on vector spaces. These are not something the vector space structure

itself provides.

In this course, we are going to give our vector spaces some additional structure.

For the first half of the course, we will grant our vector space a norm. This

allows us to assign a “length” to each vector. With this, we can easily define

convergence and continuity. It turns out this allows us to understand a lot about,

say, function spaces and sequence spaces.

In the second half, we will grant a stronger notion, called the inner product.

Among many things, this allows us to define orthogonality of the elements of a

vector space, which is something we are familiar with from, say, IB Methods.

Most of the time, we will be focusing on infinite-dimensional vector spaces,

since finite-dimensional spaces are boring. In fact, we have a section dedicated

to proving that finite-dimensional vector spaces are boring. In particular, they

are all isomorphic to

, and most of our theorems can be proved trivially for

finite-dimensional spaces using what we already know from IB Linear Algebra.

So we will not care much about them.