Weakly null continuous functions converge in L1 ($f_n\rightharpoonup 0\text{ in }C[0,1]⟹\Vert f_n{\Vert }_1\to 0$)

Weak convergence to zero gives pointwise convergence by the continuous evaluation functionals and uniform boundedness in the supremum norm by the weakly bounded set criterion. The dominated convergence theorem for Lebesgue measure then proves the displayed implication. The finite measure of the interval and uniform norm bound are the relevant hypotheses.