Dual of l spaces

Want to find the dual space of .
For each , have with .

Proposition

Norm of is .

Proof

Have (Hölder inequality)

Given let (the obvious choice, because we want it to converge in )

So so with

Have
( because )

So
Hence

Theorem

The map is an isometric isomorphism (ie they are the same spaces)

Proof

linear, as
isometry as (so is injective) (???)
Need surjective
Given let each 1
want and
For some fixed , let:

Then with
Also
ie so with as is arbitrary
Finally on for all
so on (notation for linear span)
is linear and continuous
ie on the whole of .