# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## On the uniqueness question for Hahn-Banach extensions from the space of $\mathcal {L}^1$ analytic functionsHTML articles powered by AMS MathViewer

by Edgar Reich
Proc. Amer. Math. Soc. 88 (1983), 305-310 Request permission

## Abstract:

Let $\Omega$ be a region of the complex plane, $\mathcal {B}\left ( \Omega \right )$ the space of analytic ${\mathcal {L}^1}$ functions over $\Omega$, and $\kappa \in {\mathcal {L}^\infty }\left ( \Omega \right )$. An evident necessary condition for the linear functional ${\Lambda _\kappa }\left [ \varphi \right ] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal {B}(\Omega ))$ to have a unique Hahn-Banach extension from $\mathcal {B}(\Omega )$ to ${\mathcal {L}^1}(\Omega )$ is that $|| {{\Lambda _{\kappa |G}}} || = {|| {\kappa |G} ||_\infty }$ for every restriction $\kappa |G$ of $\kappa$ to a subregion $G$ of $\Omega$. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether $\left | {\kappa (z)} \right |$ is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.
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