 Weak-strong product convergence lemma

ID: weak-strong-product-convergence-lemma

f_{n} ⇀ f

L^{p}

g_{n} \to g

L^{q}

, and

1/ p + 1/ q = 1

, then

\int f_{n} g_{n} ⟶ \int f g .

Indeed, split the difference into the weak pairing with

g

and a term bounded by the Holder inequality using

g_{n} - g

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