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 Metric 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