\'Arp\'ad Sz\'az
Linear extensions of relations between vector spaces

Comment.Math.Univ.Carolinae 44,2 (2003) 367-385.

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 $\lambda \in K\setminus \{0\}$ and $x,y\in X$. \par 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. \par 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.

Keywords: vector spaces, linear and affine subspaces, linear relations
AMS Subject Classification: Primary 26E25; Secondary 15A03, 15A04