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.
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 .
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.
Normed vector spaces are metric spaces.
It suffices to show that satisfies the metric axioms. Let
1. holds by definition and as required.
3. so the triangle inequality translates correctly.
Since the axioms hold, we conclude that V is a metric space.