= Weakly null continuous functions converge in L1
{title2=$f_n\rightharpoonup0\text{ in }C[0,1]\Longrightarrow\|f_n\|_1\to0$}
<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.
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