| 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:
|
http://hdl.handle.net/10338.dmlcz/119393 |
| . |
| Reference:
|
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| . |
