Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group G L ( 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 ∈ Σ ( X ) is an orbit of a hypercyclic operator on X . Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. w that for a separable infinite dimensional Fréchet space X , G L ( 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.