In Jarchow's *Locally Convex Spaces* this *not being quasi-complete* is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed unit ball is weakly (sequentially/countably) compact.

I see how the compactness property implies the quasi-completeness, but I do not see how the converse (that is needed here) would be deduced.

Is the asserted non-quasi-completeness "well-known", and is there a reference containing a proof, or is Jarchow's assertion false?

**Added.** (25.6.2017) Having read certain places in R. E. Edwards' *Functional Analysis*, J. Horváth's *Topological Vector Spaces* and H. Jarchow's *Locally Convex Spaces*, I have now found (from my head) the following result:

If someone can give a precise reference to an explicitly formulated previously published result with *proper proof* (not any vague explanations) of a result with the same content as Proposition 4 above, I am still interested to know. If it is posted as an answer, and I can easily check it via the Internet, I will accept it.