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Consider a (Hausdorff and complete) locally convex topological vector space $V$ and a countable subset $(v_k)_{k=1}^\infty \subset V$ of non-zero vectors.

$(*)$ Under what conditions on this subset are we guaranteed the existence of a sequence of positive real numbers $(\alpha_k)_{k=1}^\infty$, such that the series $\sum_{k=1}^\infty \alpha_k v_k$ converges (that is, the sequence of its partial sums) converges to a vector $v \in V$?

The positivity of the coefficients, $\alpha_k > 0$ for each $k$, is the crucial aspect of the question.

On the one hand, if $V$ is a Banach space, the answer is always Yes. We can always choose $\alpha_k = c_k / \|v_k\|$, where $(c_k)_{k=1}^\infty$ is any summable sequence of positive numbers.

On the other hand, consider the space $c_{00}$ of finite sequences of real numbers, turned into a complete locally convex space in the usual way. The the subset $(v_k)_{k=1}^\infty$, where the $v_k$ is the sequence that is zero everywhere except the $k$-th place, is an example for which such a sequence of $(\alpha_k)_{k=1}^\infty$ does not exist. This is clear because here a necessary and sufficient condition for the convergence of $\sum_{k=1}^\infty \alpha_k v_k$ is that only finitely many $\alpha_k$ are non-zero.

So maybe I could ask the question in a slightly different way. Is there a well-studied (joint?) property of the space $V$ and the subset $(v_k)_{k=1}^\infty$ that is sufficient (hopefully also necessary) for the answer to $(*)$ to be Yes? I was hoping that the answer would be related to one of the standard theorems of functional analysis, but I haven't recognized the right one yet.

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  • $\begingroup$ I know a couple of "usual" ways to make $c_{00}$ into a locally convex space, but unless I have got my definitions completely wrong, they aren't complete... $\endgroup$ – Nate Eldredge Jul 24 '15 at 18:33
  • $\begingroup$ @NateEldredge: How about the weak* topology on the dual to the Fréchet space of all sequences? $\endgroup$ – Igor Khavkine Jul 24 '15 at 19:17
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It's true in a Fréchet space. If $\|\cdot\|_k$, $k = 0,1,2,\ldots$, is a family of seminorms that induces the topology, choose $\alpha_j = c_j/\max\{\| v_j \|_k : k \le j\}$ for a summable sequence of positive numbers $c_j$.

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