One should be aware that an atomic measure space (in particular a finite one) might be deceptively simple.

@edamondo: In addition to Jochen's hint, the classic "Vector Measures" by Diestel and Uhl is a source to look at.

You seem to say that a subspace that is closed for some lc topology is closed for the weak topology $\sigma(V,V')$ you are mentioning; but every linear subspace is closed for $\sigma(V,V')$. So what is the consequence of this line of reasoning? Maybe I'm missing something?

Further to my comment placed below erz's proof I should mention that the result is contained in Singer's book as Theorem 13.1 on page 146ff.

Singer, on page 209 of his Vol. 1, says that the "weak basis theorem'' was proved by Bessaga and Pelczynski in their paper Properties of bases in spaces of type $B_0$ (Polish. Russian, English summaries), Pr. Mat. 3, 123--142 (1959), Zbl 0097.09203, and he also mentions a paper by Gelbaum and Wilansky's ``Functional Analysis''.

What is the difference between what your asking and that $c_0$ is an $\mathscr{L}_{\infty, 1+\varepsilon}$-space? The norm-1 projection onto $\ell_\infty^n$ is for free, isn't it?

$\beta\mathbb N$ and the spectrum of $L_\infty$ are examples of extremally disconnected compact spaces (a.k.a. Stonean spaces); in such spaces the only convergent sequences are the eventually constant ones. If $C(K)$ is order complete, then $K$ is extremally disconnected and vice versa.