Banach Spaces, An Introduction

Introduction

There are two ways to look at euclidean space. It is a vector space, with lines and planes, and scaling factors, and rigid rotations, and other transformations that respect the linear structure of the space. Or it is a space with distance, and open and closed sets, and continuous functions that respect the underlying topology.

Over the past 200 years much has been written about vector spaces, and metric spaces - and they almost seem like separate branches of mathematics. But what if a space is both a vector space and a metric space? This is a banach space (biography), and it is closer to Rⁿ than a vector space or a metric space alone. In fact a finite dimensional banach space is equivalent to Rⁿ. (We'll prove this in the following pages.) However, there are infinite dimensional banach spaces that do not resemble euclidean space.

You will need a good understanding of vector spaces and metric spaces before you proceed.