# weak convergence implies boundedness.

I have these in books without proof, mostly as a corollary. I was wondering if I could get a proof.

Suppose $$\lim_{n\to \infty} \int_0^1 f_ng dx = \int_0^1 fg dx$$ for all $g\in L^2(0,1)$, where $f_n, f \in L^2(0,1)$. Then there exists a constant $K$ such that $\|f_n\|_{L^2} \leq K \lt \infty$ for all $n$.

#### Solutions Collecting From Web of "weak convergence implies boundedness."

Banach-Steinhaus tells us that a family of linear functionals on a Banach space is either unbounded on a dense $G_\delta$ set or is uniformly bounded in norm. Since this family converges weakly we have for any $g \in L^2$ that $\sup_n \langle f_n , g \rangle < \infty$ and therefore the family of functionals is uniformly bounded in norm.