Problem
Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.
Background
Let $X$ be a normed space. Then the space $X'$ of all bounded linear functionals on $X$ is called the dual space of $X$. A sequence $(x_n)\in X$ converges weakly to $x\in X$ if for every $L\in X'$ we have $$\lim_{n\to\infty} L(x_n)=L(x).$$
Attempt
For the reverse side, I think I can just use the Lebesgue Dominated Convergence theorem. I am having trouble with the forward direction:
By the Riesz representation theorem, if $L\in (C[0,1])'$, then there is a borel measure $\mu$ so that $$L(f)=\int_0^1 f_n\ \mathrm{d}\mu.$$ Thus $f_n \rightarrow 0$ weakly if and only if $\int_0^1 f_n\ \mathrm{d}\mu\rightarrow 0.$
I am not sure where to go from here. I am hoping to set it up in such a way that the Banach Steinhaus theorem applies.