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## An application of Banach-Steinhaus theorem

Functional Analysis
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### An application of Banach-Steinhaus theorem

Suppose $1 < p < \infty$ and $p,q$ are conjugate indices, i.e., $\displaystyle \frac{1}{p} + \frac{1}{q} = 1$. If $(\mathbb{R},\mu)$ be the Lebesgue measure. If the following properties hold:

(i) $\displaystyle g \in L^{q}_{\text{loc}}(\mu)$.

(ii) $\displaystyle \int_{\mathbb{R}} |fg|\,d\mu < \infty$, for all $f \in L^{p}(\mu)$.

Then apply Banach-Steinhaus (Uniform Boundedness principle) to show that $g \in L^{q}(\mu)$.