Let a topological linear space be defined by the continuity of the linear operations only. I read on an Italian language functional analysis book, which doesn't show the proof, that any locally convex, locally bounded, separable topological linear space is normable.
In Rudin's Functional Analysis, which supposes $T_1$ to hold in a topological linear space, I find that any locally convex, locally bounded topological linear space is normable.
Does it mean that $T_1$ and separability are equivalent in such spaces? I'm inclined to suppose that separable topological linear space might be used in the sense of topological linear $T_1$-space rather than topolgical linear space containing an everywhere dense countable subset, as when separable topology is found as a synonym of topology satisfying axiom $T_1$.
Thank you so much for any answer!!!