Linear extensions of relations between vector spaces
Száz, Árpád
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 367-385 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2003-01-01
Classification:  15A03,  15A04,  26E25,  46A22,  47A06 
@article{119393,
     author = {\'Arp\'ad Sz\'az},
     title = {Linear extensions of relations between vector spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {44},
     year = {2003},
     pages = {367-385},
     zbl = {1104.26305},
     mrnumber = {2026171},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119393}
}

Száz, Árpád. Linear extensions of relations between vector spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 367-385. http://gdmltest.u-ga.fr/item/119393/

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