An application of Banach-Steinhaus theorem
Posted: Thu May 18, 2017 7:58 pm
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 $g \in L^{q}(\mu)$.
(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 $g \in L^{q}(\mu)$.