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.


Sequential continuity criterion for a linear map on a metrizable topological vector space

A topological vector space is metrizable when its topology is induced by a <metric>. At the origin it then has a countable neighborhood basis, so continuity of a <linear map> can be tested on sequences.


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 Incoming links: Metrizable topological vector space