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Is there a criterion for a sequence in $C[0,1]$ to converge weakly?
Let $\{f_n\}$ be a sequence in $C[0,1]$, $f\in C[0,1]$. Suppose $f_n\to f$ (weakly). Then for each $x\in [0,1]$ $ev_x(f_n)\to ev_x(f)$, i.e. $f_n(x)\to f(x)$. Thus, pointwise convergence is necessary. I guess it is not enough. Could you give any reference?

Kolyan
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    Boundedness in the sup-Norm is also necessary (uniform boundedness principle). By the dominated convergence theorem, this also suffices (together with pointwise convergence). – PhoemueX Jun 19 '14 at 20:06

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