Normed Space

Let be a real or complex vector space with Norm
A normed space is a pair

NOTE: a normed space gives rise to a metric space with .
This then induces a Topological Space.
So we can talk about open sets, closed sets, convergent sequences etc.

L norms
Banach Space
Some properties
Unit Ball

Separable

Equivalent Norms

Open Mapping Lemma

New spaces from old

For normed spaces and can define a norm on by:

This space is written

Similarly, have with
And similarly for any .

All are equivalent and all induce the product topology on
Note that and being Banach Space is Banach.
And also that and are always closed subspaces of

Quotients
Completion