Linear extensions of relations between vector spaces

Száz, Árpád. "Linear extensions of relations between vector spaces." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 367-385. <http://eudml.org/doc/249167>.

@article{Száz2003,
abstract = {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 \lbrace 0\rbrace $ 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.},
author = {Száz, Árpád},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector spaces; linear and affine subspaces; linear relations; linear and affine subspaces; linear relations},
language = {eng},
number = {2},
pages = {367-385},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Linear extensions of relations between vector spaces},
url = {http://eudml.org/doc/249167},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Száz, Árpád
TI - Linear extensions of relations between vector spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 2
SP - 367
EP - 385
AB - 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 \lbrace 0\rbrace $ 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.
LA - eng
KW - vector spaces; linear and affine subspaces; linear relations; linear and affine subspaces; linear relations
UR - http://eudml.org/doc/249167
ER -