Topology/Normed Vector Spaces

A normed vector space is a vector space V with a function that represents the length of a vector, called a norm.

DefinitionEdit

We know the vector space defintion, so we need to define the norm function.   is a norm if these three conditions hold.

1. Only the zero vector has zero length, with all others being positive.   for all  .

2. For   and   we have  .

3. The triangle inequality holds:   for all  .

ExampleEdit

For a given   we know that   is a vector space and its norm can be defined to be   ie.  . This is not unusual, in fact we say that a norm induces a metric with the first equation. So normed vector spaces are always metric spaces. Let's prove this.

TheoremEdit

Normed vector spaces are metric spaces.

Proof

It suffices to show that   satisfies the metric axioms. Let  

1.   holds by definition and   as required.

2.  

3.   so the triangle inequality translates correctly.

Since the axioms hold, we conclude that V is a metric space.

ExercisesEdit

(under construction)