Non-uniqueness of certain Hahn--Banach extensions
Vladikavkazskij matematičeskij žurnal, Tome 6 (2004) no. 1, pp. 26-28
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Let $f$ be a continuous linear functional defined on a subspace $M$ of a normed space $X$. If $X$ is real or complex, there are results that characterize uniqueness of continuous extensions $F$ of $f$ to $X$ for every subspace $M$ and those that apply just to $M$. If $X$ is defined over a non-Archimedean valued field $K$ and the norm also satisfies the strong triangle inequality, the Hahn–Banach theorem holds for all subspaces $M$ of $X$ if and only if $K$ is spherically complete and it is well-known that Hahn–Banach extensions are never unique in this context. We give a different proof of non-uniqueness here that is interesting for its own sake and may point a direction in which further investigation would be fruitful.
@article{VMJ_2004_6_1_a3,
author = {E. Beckenstein and L. Narici},
title = {Non-uniqueness of certain {Hahn--Banach} extensions},
journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
pages = {26--28},
publisher = {mathdoc},
volume = {6},
number = {1},
year = {2004},
language = {en},
url = {http://geodesic.mathdoc.fr/item/VMJ_2004_6_1_a3/}
}
E. Beckenstein; L. Narici. Non-uniqueness of certain Hahn--Banach extensions. Vladikavkazskij matematičeskij žurnal, Tome 6 (2004) no. 1, pp. 26-28. http://geodesic.mathdoc.fr/item/VMJ_2004_6_1_a3/