= Original and weak continuity of linear maps between Fréchet spaces
{title2=$T:(E,\tau_E)\to(F,\tau_F)$}
For <Fréchet spaces>, the two continuity conditions for a <linear map> are equivalent. Original continuity makes every target <continuous linear functional> compose to a source <continuous linear functional>, proving weak continuity. Conversely, weak continuity makes these compositions continuous in the original source <topology>. They separate target points, so limits in the <graph of a linear operator> remain in the graph. The <closed graph theorem for Fréchet spaces> finishes the proof.
Without the hypotheses, equal <continuous dual spaces> do not imply equal <topologies>. The identity from an infinite-dimensional <Hilbert space> with its <weak topology> to its norm <topology> is weak-to-weak continuous but not continuous in those original <topologies>.
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