# Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?

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.

• Have managed to dig up Grotndieck's text "Espaces vectoriels topologiques". The result mentioned is on p. 277. By the way, if $E$ is a Banach space, then it bidual is the linear span of the bipolar of its unit ball in the algebraic dual of its dual. The latter is, by the bipolar theorem, its closure in the weak topology induced by $E'$. From this, everything follows. – traun Jun 24 '17 at 6:04