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 .