Quotients

For Normed Space, a closed subspace of , have Quotient vector space with projection map .
Want to put a norm on

Proposition

normed, a closed subspace of .
Then defines a norm on . Moreover, is continuous
is Banach Space is Banach.

Proof

a norm:

Need that and
In other words that for all in we have
Have with and
Knowing , we conclude that

continuous

Have for all (definition of )

Banach Banach

For all with there is some with and , so is dense in
Done by Open Mapping Lemma

WARNING If continuous, then it is not always true that isomorphic to the image.
EG identity from to : is complete but image isn’t.