= Sequential continuity criterion in a metrizable vector space
A linear map from a <metrizable space> carrying a vector-space topology is continuous if it takes every null sequence to a null sequence. If continuity fails, choose a point in each shrinking basic neighborhood whose image stays outside a fixed neighborhood of zero. These points converge to zero but their images do not. For a <Fréchet space> described by increasing <seminorms> $p_j$, a discontinuous functional admits $p_j(f_j)\leq1/j$ and $|u(f_j)|\geq1$.
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