{$g_n$}$_{n \in\mathbb{N}}$ converges to $g$ in $L^q(\mathbb{R^n})$ then {$A_f(g_n)$}$_{n \in \mathbb{N}}$ in $\mathbb{C}$ converges to $A_f(g)$.

Question

Let $1<p,q<\infty$ with $\dfrac{1}{p}+\dfrac{1}{q}=1$ and $f \in L^p(\mathbb{R^n})$. Define the functional $A_f: L^q(\mathbb{R^n}) \to \mathbb{C}$ by:

$$A_f(g)=\int_{\mathbb{R^n}}f(x)g(x)dx$$

I must show that: If a sequence {$g_n$}$_{n \in\mathbb{N}}$ converges to $g$ in $L^q(\mathbb{R^n})$ then {$A_f(g_n)$}$_{n \in \mathbb{N}}$ in $\mathbb{C}$ converges to $A_f(g)$.

What I have tried:

$\left \| g_n-g \right \|_q \to 0, n \to \infty$

$\left | A_f(g_n)-A_f(g) \right |=\left |\int_{\mathbb{R^n}}f(x)g_n(x) dx - \int_{\mathbb{R^n}}f(x)g(x) dx\right |=\left |\int_{\mathbb{R^n}}f(x)(g_n(x)-g(x)) dx\right |\leq \int_{\mathbb{R^n}}\left |f(x)\right | \left |g_n(x)-g(x)\right | dx$

Then I would use the assumption that $\left \| g_n-g \right \|_q \to 0, n \to \infty$ and thus that would be 0.

Is this right?

Answer 1

Hint: Use Holder inequality to show that $A_f$ is bounded