Sequential continuity criterion for a linear map on a metrizable topological vector space
= Sequential continuity criterion for a linear map on a metrizable topological vector space
A <linear map> from a <metrizable topological vector space> is continuous exactly when it maps every sequence converging to zero to a sequence converging to zero. If continuity fails, a decreasing countable neighborhood basis supplies a null sequence whose images stay outside one fixed neighborhood of zero.