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Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group $GL(X)$ of invertible linear operators on a separable infinite dimensional Banach space $X$ acts transitively on the set $Σ(X)$ of countable dense linearly independent subsets of $X$. As a consequence, each $A\in Σ(X)$ is an orbit of a hypercyclic operator on $X$. Furthermore, every countably dimensional normed space supports a hypercyclic operator. We show that for a separable infinite dimensional Fréchet space $X$, $GL(X)$ acts transitively on $Σ(X)$ if and only if $X$ possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.