where .
Example
using and .
Linear Analysis (old)
For , the conjugate index to is the with
Lemma
Let be conjugate and . Then .
Proof
WLOG . Put .
So need to show that .
Take and then it’s immediate from concavity of .
Hölder’s inequality
Let , be conjugate and .
Then with (see L norms).
Proof
WLOG have .
Now for every have (due to lemma).
Now sum and get .