Problem about weak sequential continuity of a nonlinear map from $L^4(\mathbb R^N)$ to $\mathbb R$

Question

Let's consider $f\in L^2(\mathbb R^N)$ such that $f(x)>0$ for all $x\in \mathbb R^N$. Consider the nonlinear functional $\phi:L^4(\mathbb R^N) \to \mathbb R$ given by $$\phi(g) = \int_{\mathbb R^N} f(x) |g(x)|^2 dx.$$

I was able to prove that $\phi$ is continuous with respect to the strong topology and is not continuous when we use $L^4(\mathbb R^N)$ with the weak topology.

How do i prove that $\phi$ is weakly sequentially continuous, that is, if $g_n$ is an sequence in $L^4(\mathbb R^N)$ that converges to $g\in L^4(\mathbb R^N)$ in the weak topology how do i show that $\phi(g_n) \to \phi(g)$?

Answer 1

You do not.

You need convergence almost everywhere to obtain the result, otherwise a counterexample can be found. For instance, consider

$$g_n(x)=\sin(nx), \, x \in [-\pi,\pi]$$.

It is weakly convergent to zero, but $\phi(g_n) \nrightarrow 0$.