2 - Normed Linear Spaces

Summary

2.1 Definitions and Examples

We now introduce the fundamental object in our investigations. A vector space, while very useful, is somewhat unwieldy when it is infinite dimensional. Equipping it with a metric, especially one that understands the linear structure of the underlying set, is a simple and effective way to alleviate this problem.

2.1.1 DEFINITION A norm on a K-vector space E is a function

which satisfies the following properties for all x, y ∊ E and all α ∊ K.