Zabreiko's Lemma

Question

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.

Answer 1

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

• t.b.'s answer to the question Direct approach to the Closed Graph Theorem contains a proof of this result.
• The blog post Zabreiko’s lemma and four fundamental theorems of functional analysis was based on the above answer.

Papers

• П. П. Забрейко, Об одной теореме для полуаддитивных функционалов, Функциональный анализ и его приложения, 3:1 (1969), 86–88.
• P. P. Zabreiko, A theorem for semiadditive functionals, Functional analysis and its applications 3 (1), 1969, 70-72). MR 241947, ZentralBlatt 0199.21202.

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.

Answer 2

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.

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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 $c\ge 0$.