Unit Ball

Let be a Normed Space.
The unit ball of is:

Can check that is closed, bounded, convex, symmetric and a neighbourhood of 0.

Lemma

Let be a Vector Space over or .
If is closed, bounded, convex, symmetric and a Neighbourhood of
then defines a Norm by taking:

Furthermore, is the unit ball in this norm.

Theorem

Let be an -dimensional Normed Space.
Then there are some with for all and for all .
In particular, is not Complete.