Linear extensions of relations between vector spaces

Árpád Száz

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 2, page 367-385
  • ISSN: 0010-2628

Abstract

top
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 λ F ( x ) F ( λ x ) and F ( x ) + F ( y ) F ( x + y ) for all λ K { 0 } and x , y X . After improving and supplementing some former results on linear relations, we show that a relation Φ 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 Φ ( e ) Y | Z for all e 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 Ψ of F | Z can be extended to a linear selection relation Φ 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 F - 1 is also a function.

How to cite

top

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 -

References

top
  1. Adasch N., Der Satz über offene lineare Relationen in topologischen Vektorräumen, Note Mat. 11 (1991), 1-5. (1991) MR1258535
  2. Arens R., 10.2140/pjm.1961.11.9, Pacific J. Math. 11 (1961), 9-23. (1961) Zbl0102.10201MR0123188DOI10.2140/pjm.1961.11.9
  3. Berge C., Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity, Oliver and Boyd London (1963). (1963) Zbl0114.38602MR1464690
  4. Cross R., Multivalued Linear Operators, Marcel Dekker New York (1998). (1998) Zbl0911.47002MR1631548
  5. Dacić R., On multi-valued functions, Publ. Inst. Math. (Beograd) (N.S.) 9 (1969), 5-7. (1969) MR0257991
  6. Findlay G.D., 10.4153/CMB-1960-015-x, Canad. Math. Bull. 3 (1960), 131-132. (1960) Zbl0100.28002MR0124251DOI10.4153/CMB-1960-015-x
  7. Godini G., Set-valued Cauchy functional equation, Rev. Roumaine Math. Pures Appl. 20 (1975), 1113-1121. (1975) Zbl0322.39013MR0393920
  8. Holá L'., Some properties of almost continuous linear relations, Acta Math. Univ. Comenian. 50-51 (1987), 61-69. (1987) MR0989404
  9. Holá L'., Kupka I., Closed graph and open mapping theorems for linear relations, Acta Math. Univ. Comenian. 46-47 (1985), 157-162. (1985) MR0872338
  10. Holá L'., Maličký P., Continuous linear selectors of linear relations, Acta Math. Univ. Comenian. 48-49 (1986), 153-157. (1986) MR0885328
  11. Kelley J.L., Namioka I., Linear Topological Spaces, D. Van Nostrand New York (1963). (1963) Zbl0115.09902MR0166578
  12. Nikodem K., K-convex and K-concave set-valued functions, Zeszty Nauk. Politech. Lódz. Mat. 559 (1989), 1-75. (1989) 
  13. Robinson S.M., 10.1090/S0002-9947-1972-0313769-9, Trans. Amer. Math. Soc. 174 (1972), 127-140. (1972) MR0313769DOI10.1090/S0002-9947-1972-0313769-9
  14. Smajdor W., Subadditive and subquadratic set-valued functions, Prace Nauk. Univ. Ślask. Katowic. 889 (1987), 1-73. (1987) Zbl0626.54019MR0883802
  15. Száz Á., Pointwise limits of nets of multilinear maps, Acta Sci. Math. (Szeged) 55 (1991), 103-117. (1991) MR1124949
  16. Száz Á., 10.4064/ap-69-3-235-249, Ann. Polon. Math. 69 (1998), 235-249. (1998) MR1665007DOI10.4064/ap-69-3-235-249
  17. Száz Á., An extension of Kelley's closed relation theorem to relator spaces, Filomat (Nis) 14 (2000), 49-71. (2000) Zbl1012.54026MR1953994
  18. Száz Á., Preseminorm generating relations and their Minkowski functionals, Publ. Elektrotehn. Fak. Univ. Beograd, Ser. Mat. 12 (2001), 16-34. (2001) Zbl1060.46004MR1920353
  19. Száz Á., Translation relations, the building blocks of compatible relators, Math. Montisnigri, to appear. MR2023739
  20. Száz Á., Száz G., Additive relations, Publ. Math. Debrecen 20 (1973), 259-272. (1973) MR0340878
  21. Száz Á., Száz G., Linear relations, Publ. Math. Debrecen 27 (1980), 219-227. (1980) MR0603995
  22. Ursescu C., Multifunctions with convex closed graph, Czechoslovak Math. J. 25 (1975), 438-441. (1975) Zbl0318.46006MR0388032
  23. Wilhelm M., 10.4064/fm-114-3-219-228, Fund. Math. 114 (1981), 219-228. (1981) MR0644407DOI10.4064/fm-114-3-219-228

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.