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.