Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p: X \to [0,\infty)$ be a seminorm. If for all absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leq \sum_{n=1}^\infty p(x_n) \in [0,\infty] $$ then $p$ is continuous.
I must find this lemma's proof.
I'll post a CW answer here so that the question does not remain unanswered (it was basically answered in comments). And this answer can also be used to collect some further references. (The answer is community wiki, so do not hesitate to edit it if you have something to add.)
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Zabreiko's lemma does not only imply the open mapping theorem and has a quite similar proof, it also easily follows from the open mapping theorem.
Indeed, $q(x)=\|x\|+p(x)$ is a norm on $X$ such that the identity $i:(X,q)\to (X,\|\cdot\|)$ is obviously continuous. If $(X,q)$ is complete then $i^{-1}$ is continuous which implies the continuity of $p$.
The completeness of $(X,q)$ follows from the simple and well-know fact that a normed space is complete if every absolutely convergent series converges:
Let thus $\sum x_n$ be a series with $\sum q(x_n)<\infty$. Then $\sum \|x_n\|<\infty$ and the series $s=\sum x_n$ converges in $(X,\|\cdot\|)$. To show that it also converges in $(X,q)$ take $n\in\mathbb N$ and estimate (here the assumption of Zabreiko's lemma is used) $$q(s-\sum_{k=1}^nx_k)=q(\sum_{k=n+1}^\infty x_k)\le \sum_{k=n+1}^\infty q(x_k)\to 0$$ since the series $\sum q(x_k)$ converges.
These arguments also apply in the case of a Fréchet space $X$ with an increasing sequence of semi-norms $\|\cdot\|_\ell$ generating the topology. Here, absolute convergence of a series means $\sum_n\|x_n\|_\ell<\infty$ for every $\ell$. The continuity of $p$ yields $p\le c \|\cdot\|_\ell$ for some $\ell\in\mathbb N$ and $c\ge 0$.
