Notes on Cambridge Maths

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Hölder inequality

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 .

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