L norms

On vectors

Norms for any defined by

On sequences

Let be the space of all scalar sequences. We define the normed space

With norm .

These all satisfy the desired properties by Minkowski’s Inequality.

On functions

Write (or just for the whole ) for the space of all functions such that converges.
Then the norm is defined to be:

Dual of l spaces

Spanning

Note that do not span
but they do span the space of finite sequences .
We do get close because is the closure of for .

This is FALSE for , e.g. let .

Also subspaces need not be closed - e.g. in .
Even have dense in .