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