Is a Frechet space separable, if its dual is?

It is known, and not too difficult to prove that if a dual of a normed space is separable, it is separable itself. Does the same hold for Frechet spaces?

(By the dual I mean the dual endowed with the strong topology.)

• Did you try to transfer the usual proof in case of a normed vector space? – gerw Apr 9 '18 at 6:49
• @gerw The proof that I know relies on the norm heavily: you should be able to choose "small" functionals and "large" vectors. If you know a transferable proof, could you please give a reference? – erz Apr 9 '18 at 7:02
• Do you consider the weak*-topology or the strong topology on the dual space? Note that $F^*$ with the strong dual is in general not metrizable. In fact, this happens only if $F$ is already normable, i.e. a banach space. In this situation we can apply the result for banach spaces. – p4sch Apr 9 '18 at 8:39
• @p4sch I mean the strong topology. I know that $F^*$ is not always metrizable, but in my specific case it is separable, and I want to conclude that $F$ is then separable too, without assuming that $F$ is normed. – erz Apr 9 '18 at 8:48