A hypercyclicity criterion for non-metrizable topological vector spaces

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.