$f \mapsto \int f^2$ is $L^1$-weakly lower semicontinuous

Question

If $f \in C(\mathbb [0,1], \mathbb R)$ is $$ f \mapsto \int_0^1 f(t)^2\ dt$$ $L^1$-weakly lower semicontinuous? I.e. if $$\int_0^1 f_n g \rightarrow \int_0^1 f g$$ for every $g \in L^{\infty}$, then $$ \int_0^1 f^2 \leq \liminf_n \int_0^1 f_n^2 \qquad ? $$