  We say that a (countably dimensional) topological vector space $X$ is orbital
if there is $T\in L(X)$ and a vector $x\in X$ such that $X$ is the linear span
of the orbit ${T^nx:n=0,1,...}$. We say that $X$ is strongly orbital if,
additionally, $x$ can be chosen to be a hypercyclic vector for $T$. Of course,
$X$ can be orbital only if the algebraic dimension of $X$ 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 $X$ does not have the invariant subset property. That is, there is
$T\in L(X)$ such that every non-zero $x\in X$ is a hypercyclic vector for $T$.
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 Fr\'echet space $X$. For instance, in
$X=\ell_2\times \omega$, there are exactly 3 pairwise non-isomorphic (as
topological vector spaces) dense countably dimensional subspaces.
