  We provide a sufficient condition for an operator $T$ on a non-metrizable and
sequentially separable topological vector space $X$ to be sequentially
hypercyclic. This condition is applied to some particular examples, namely, a
composition operator on the space of real analytic functions on $]0,1[$, which
solves two problems of Bonet and Doma\'nski \cite{bd12}, and the "snake shift"
constructed in \cite{bfpw} on direct sums of sequence spaces. The two examples
have in common that they do not admit a densely embedded F-space $Y$ for which
the operator restricted to $Y$ is continuous and hypercyclic, i.e., the
hypercyclicity of these operators cannot be a consequence of the comparison
principle with hypercyclic operators on F-spaces.
