# Article

 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: [1] Adasch N.: Der Satz über offene lineare Relationen in topologischen Vektorräumen.Note Mat. 11 (1991), 1-5. MR 1258535 Reference: [2] Arens R.: Operational calculus of linear relations.Pacific J. Math. 11 (1961), 9-23. Zbl 0102.10201, MR 0123188, 10.2140/pjm.1961.11.9 Reference: [3] Berge C.: Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity.Oliver and Boyd London (1963). Zbl 0114.38602, MR 1464690 Reference: [4] Cross R.: Multivalued Linear Operators.Marcel Dekker New York (1998). Zbl 0911.47002, MR 1631548 Reference: [5] Dacić R.: On multi-valued functions.Publ. Inst. Math. (Beograd) (N.S.) 9 (1969), 5-7. MR 0257991 Reference: [6] Findlay G.D.: Reflexive homomorphic relations.Canad. Math. Bull. 3 (1960), 131-132. Zbl 0100.28002, MR 0124251, 10.4153/CMB-1960-015-x Reference: [7] Godini G.: Set-valued Cauchy functional equation.Rev. Roumaine Math. Pures Appl. 20 (1975), 1113-1121. Zbl 0322.39013, MR 0393920 Reference: [8] Holá L'.: Some properties of almost continuous linear relations.Acta Math. Univ. Comenian. 50-51 (1987), 61-69. MR 0989404 Reference: [9] Holá L'., Kupka I.: Closed graph and open mapping theorems for linear relations.Acta Math. Univ. Comenian. 46-47 (1985), 157-162. MR 0872338 Reference: [10] Holá L'., Maličký P.: Continuous linear selectors of linear relations.Acta Math. Univ. Comenian. 48-49 (1986), 153-157. MR 0885328 Reference: [11] Kelley J.L., Namioka I.: Linear Topological Spaces.D. Van Nostrand New York (1963). Zbl 0115.09902, MR 0166578 Reference: [12] Nikodem K.: K-convex and K-concave set-valued functions.Zeszty Nauk. Politech. Lódz. Mat. 559 (1989), 1-75. Reference: [13] Robinson S.M.: Normed convex processes.Trans. Amer. Math. Soc. 174 (1972), 127-140. MR 0313769, 10.1090/S0002-9947-1972-0313769-9 Reference: [14] Smajdor W.: Subadditive and subquadratic set-valued functions.Prace Nauk. Univ. Ślask. Katowic. 889 (1987), 1-73. Zbl 0626.54019, MR 0883802 Reference: [15] Száz Á.: Pointwise limits of nets of multilinear maps.Acta Sci. Math. (Szeged) 55 (1991), 103-117. MR 1124949 Reference: [16] Száz Á.: The intersection convolution of relations and the Hahn-Banach type theorems.Ann. Polon. Math. 69 (1998), 235-249. MR 1665007, 10.4064/ap-69-3-235-249 Reference: [17] Száz Á.: An extension of Kelley's closed relation theorem to relator spaces.Filomat (Nis) 14 (2000), 49-71. Zbl 1012.54026, MR 1953994 Reference: [18] Száz Á.: Preseminorm generating relations and their Minkowski functionals.Publ. Elektrotehn. Fak. Univ. Beograd, Ser. Mat. 12 (2001), 16-34. Zbl 1060.46004, MR 1920353 Reference: [19] Száz Á.: Translation relations, the building blocks of compatible relators.Math. Montisnigri, to appear. MR 2023739 Reference: [20] Száz Á., Száz G.: Additive relations.Publ. Math. Debrecen 20 (1973), 259-272. MR 0340878 Reference: [21] Száz Á., Száz G.: Linear relations.Publ. Math. Debrecen 27 (1980), 219-227. MR 0603995 Reference: [22] Ursescu C.: Multifunctions with convex closed graph.Czechoslovak Math. J. 25 (1975), 438-441. Zbl 0318.46006, MR 0388032 Reference: [23] Wilhelm M.: Criteria of openness of relations.Fund. Math. 114 (1981), 219-228. MR 0644407, 10.4064/fm-114-3-219-228 .

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