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 $\mathcal{L}_{2}[a,b]$