# Zabreiko's Lemma

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.

• Robert E. Megginson's An Introduction to Banach Space Theory has one on page 42. (The statement there requires the inequality hold for convergent series.) – David Mitra Jun 21 '14 at 18:45
• It seems a proof of this is contained in this post. – David Mitra Jun 21 '14 at 18:59
• I want to advertise the most recent blog post on our site. More references in there. – Jyrki Lahtonen Jun 29 '14 at 8:06