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Title: Linear extensions of relations between vector spaces (English)
Author: Száz, Árpád
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 44
Issue: 2
Year: 2003
Pages: 367-385
Category: math
Summary: Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F(x)\subset F(\lambda x)$ and $F(x)+F(y)\subset F(x+y)$ for all $\lambda \in K\setminus \{0\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi (e)\in Y|Z$ for all $e\in E$. Moreover, if $E$ generates $X$, then this extension is unique. Furthermore, we also prove that if $F$ is a linear relation of $X$ into $Y$ and $Z$ is a linear subspace of $X$, then each linear selection relation $\Psi$ of $F|Z$ can be extended to a linear selection relation $\Phi$ of $F$. A particular case of this Hahn-Banach type theorem yields an easy proof of the existence of a linear selection function $f$ of $F$ such that $f\circ F^{ -1}$ is also a function. (English)
Keyword: vector spaces
Keyword: linear and affine subspaces
Keyword: linear relations
MSC: 15A03
MSC: 15A04
MSC: 26E25
MSC: 46A22
MSC: 47A06
idZBL: Zbl 1104.26305
idMR: MR2026171
Date available: 2009-01-08T19:29:53Z
Last updated: 2020-02-20
Stable URL:
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