Orbital and strongly orbital spaces

We say that a (countably dimensional) topological vector space XX is orbital if there is T∈L(X)T∈L(X) and a vector x∈Xx∈X such that XX is the linear span of the orbit {Tnx:n=0,1,…}{Tnx:n=0,1,…}. We say that XX is strongly orbital if, additionally, xx can be chosen to be a hypercyclic vector for TT. Of course, XX can be orbital only if the algebraic dimension of XX is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space XX does not have the invariant subset property. That is, there is T∈L(X)T∈L(X) such that every non-zero x∈Xx∈X is a hypercyclic vector for TT. Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space.As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fréchet space XX. For instance, in X=ℓ2×ωX=ℓ2×ω, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.