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Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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). (MathSciNet review.)

Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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). (MathSciNet review.)

Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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).

Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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). (MathSciNet review.)

Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geq 0$ such that $p(x)\leq C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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).

Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have $$ p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty]. $$ Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.

Original references:

• 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).