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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.

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  • $\begingroup$ 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.) $\endgroup$ – David Mitra Jun 21 '14 at 18:45
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    $\begingroup$ It seems a proof of this is contained in this post. $\endgroup$ – David Mitra Jun 21 '14 at 18:59
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    $\begingroup$ I want to advertise the most recent blog post on our site. More references in there. $\endgroup$ – Jyrki Lahtonen Jun 29 '14 at 8:06
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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.)

On this site

Papers

Books

  • The book Robert E. Megginson: An Introduction to Banach Space Theory, Springer, New York, 1998 (GTM 183) contains this result with proof in Lemma 1.6.3
  • The article Lemma von Zabreiko (current revision) on German Wikipedia also lists this book among references: Vasile I. Istrățescu: Strict convexity and complex strict convexity, Lecture Notes in Pure and Applied Mathematics, Band 89, Marcel Dekker (1984), ISBN 0-8247-1796-1. I do not have access to this book, but I see that this result also appears in another book by the same author - Inner Product Structures: Theory and Applications - as Lemma 2.5.3.
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