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Lets say that $f:[a,b] \rightarrow \mathbb{R}$ is a measurable function such that $H: \mathcal{L}_{2}[a,b] \rightarrow \mathbb{R}$ defined as

$H(g) = \int_{a}^{b}fg$

is finite for all $g \in \mathcal{L}_{2}[a,b]$

I was wondering if $H$ is a bounded linear functional on $\mathcal{L}_{2}[a,b]$

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    Hardy-Littlewood-Polya's book *Inequalities* calls this result "the converse of Hölder's inequality". Cfr. Theorem 190 (161 for the infinite sum and 15 for the finite sum case). More precisely, Theorem 190 states that $(H(g)\ \text{is finite for all }g\in L^2[a, b] )\Rightarrow (f \in L^2[a, b])$, from which continuity of $H$ follows easily. This is exactly what Davide has done below.2012-04-11

2 Answers 2