On n-collinear elements and Riesz theorem

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1 Available olie at J. Noliear Sci. Appl. 9 (206), Research Article O -colliear elemets ad Riesz theorem Wasfi Shataawi a, Mihai Postolache b, a Departmet of Mathematics, Hashemite Uiversity, P. O. Box 50459, Zarqa 35, Jorda. b Departmet of Mathematics ad Iformatics, Uiversity Politehica of Bucharest, Bucharest, , Romaia. Commuicated by R. Saadati Abstract I this paper, we prove that the -colliear elemets x, x 2,..., x, u satisfy some special relatios i a -ormed space X. Further, we prove that u x + +x is the oly uique elemet i the -ormed space X such that x, x 2,..., x, u are -colliear elemets i X satisfyig some specified iequalities. Moreover, we prove that the Riesz theorem holds whe X is a liear -ormed space. c 206 All rights reserved. Keywords: -ormed space, 2-ormed spaces, liearly depedet, colliear elemets. 200 MSC: 5A39.. Itroductio Misiak [0, ] defied -ormed spaces ad ivestigated the properties of these spaces. The cocept of a -ormed space is a geeralizatio of the cocepts of a ormed space ad of a 2-ormed space. Let X ad Y be metric spaces. A mappig f : X Y is called a isometry if f satisfies d Y (fx, fy) d X (x, y) for all x, y X, where d X (, ) ad d Y (, ) deote the metrics i the spaces X ad Y, respectively. For some fixed umber r > 0, suppose that f preserves distace r; that is, for all x, y i X with d X (x, y) r, we have d Y (fx, fy) r. The r is called a coservative (or preserved) distace for the mappig f. The basic problem of coservative distaces is whether the existece of a sigle coservative distace for some f implies that f is a isometry of X ito Y. It is called the Aleksadrov problem. The Aleksadrov problem has bee extesively studied by may authors (see [, 6, 7, 9, 2, 3]). I 2004, Chu et al. [7] defied Correspodig author addresses: swasfi@hu.edu.jo (Wasfi Shataawi), mihai@mathem.pub.ro (Mihai Postolache) Received

2 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), the cocept of -isometry which is suitable for represetig the otio of -distace preservig mappigs i liear -ormed spaces ad studied the Aleksadrov problem i liear -ormed spaces. For related works we refer the reader to [2, 3, 4, 5, 7, 8]. The cocept of -colliear elemets i the -ormed space X plays a major rule i coservative distace, for this reaso the authors studies some special relatios i the -ormed space X. 2. Basic Cocepts Defiitio 2. ([]). Let X be a real liear space with dim X ad,, : } X X {{ X } R times be a fuctio. The (X,,, ) is called a liear -ormed space if. x,..., x 0 iff x,..., x are liearly depedet; 2. x,..., x x j,..., x j for ay permutatio (j, j2,..., j) of (, 2,..., ); 3. βx,..., x β x,..., x ; 4. x + y, x 2,..., x x, x 2,..., x + y, x 2,..., x for all β R ad x, y, x,..., x X. The fuctio,, is called a -orm o X. Defiitio 2.2 ([4]). The poits x 0, x,..., x of X are said to be -colliear if for every i, the set {x j x i : 0 j i } is liearly depedet. Remark 2.3. If the poits x 0, x,..., x of X are -colliear, the there are scalars λ 0, λ,..., λ ot all 0 such that x i0 λ ix i i0 λ. i Followig the Defiitio 3.2 of [5] of 2-closed sets i 2-ormed space X, we itroduce the followig defiitio. Defiitio 2.4. Let W be a subset of a -ormed space X. The W is called a -closed set if for x, x 2,..., x X such that if x w, x 2 w,..., x w 0, w W the there is w 0 W such that x w 0, x 2 w 0,..., x w 0 0. From ow o, uless otherwise stated, we let X be a liear -ormed space with dim(x) Mai Results We start our works by provig the followig propositio. Propositio 3.. Give x,..., x X. Let u t x + t 2 x t x t + t t for some scalars t, t 2,..., t ot all are 0. The u satisfies the followig relatios:

3 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), x c, x 2 c,..., x j c, x j u, x j+ c,..., x c j {2, 3,..., }, 2. x u, x 2 c, x 3 c,..., x c i2 t i ad 3. x c, x 2 c,..., x c, x u i t i x c, x 2 c,..., x c 0. t j i t i i t i x c, x 2 c,..., x c, x c,..., x c, for all i t i x c, x 2 c,..., x c for some c X with Proof. To prove, choose c X with x c, x 2 c,..., x c 0. Give j {2,..., }. The x c,..., x j c, x j u, x j+ c,..., x c x i c,..., x j c, x j t ix i i t, x j+ c,..., x c i i t i x c,..., x j c, t i x i + t i x j, x j+ c,..., x c. Let The Let The w t c t c + t 2 c t 2 c t j c t j c + t j+ c t j+ c t c t c. t i x i + t i x j v x c,..., x j c, x j u, x j+ c,..., x i t i x c,..., x j c, v + t i x i + t i (c x i ) + t i (c x i ). t i t i x j + w t i (x j c). (x j c), x j+ c,..., x c. Sice c x, c x 2,..., c x j, c x j+,..., c x, v are liearly depedet, we have x c,..., x j c, x j u, x j+ c,..., x c i t i x c,..., x j c, t i (x c), x j+ c,..., x c t i i t x c, x 2 c,..., x c. i By the same argumet we ca prove 2 ad 3.

4 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), The followig remark is a direct applicatio to Propositio 3.. Remark 3.2. Let x, x 2,, x be elemets i the -ormed space X. The satisfies the followig equalities: u x + x x. x c, x 2 c,..., x j c, x j u, x j+ c,..., x c x c,..., x c for all j {2, 3,..., }, 2. x u, x 2 c, x 3 c,..., x c x c, x 2 c,..., x c, ad 3. x c, x 2 c,..., x c, x u x c, x 2 c,..., x c for some c X with x c, x 2 c,..., x c 0. Propositio 3.3. Give x,..., x X. Let u t x + t 2 x t x t + t t for some scalars t, t 2,, t ot all are 0. The u satisfies the followig relatios:. x u, x c,..., x j c, x j+ c,..., x c t j i t i x c,..., x c, for all j {2, 3,..., }, 2. x 2 u, x 2 c, x 3 c,..., x c t i t i x c, x 2 c,..., x c, ad 3. x u, x c, x 2 c,..., x c t i t i x c, x 2 c,..., x c for some c X with x c, x 2 c,..., x c 0. Proof. Choose c X with x c, x 2 c,..., x c 0. Give j {2, 3,..., }. The x u, x c,..., x j c, x j+ c,..., x c x t x + + t x, x c,..., x j c, x j+ c,..., x c t + + t t + + t (t t )x (t 2 x t x ), x c,..., x j c, x j+ c,..., x c. Let w t 2 c t 2 c + t 3 c t 3 c + + t c t c. The (t t )x (t 2 x t x ) (t t )x (t 2 x t x ) + w (t t )x + t 2 (c x 2 ) + + t (c x ) c(t t ) (t t )(x c) t 2 (x 2 c) t (x c). Let v (t t )(x c) t i (x i c). i2,i j

5 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), The x u, x c,..., x j c, x j+ c,..., x c t + + t v t j(x j c), x c,..., x j c, x j+ c,..., x c. Sice x c, x 2 c,..., x j c, x j+ c,..., x c, v are liearly depedet, we have x u, x c,..., x j c, x j+ c,..., x c t + + t t j(x j c), x c,..., x j c, x j+ c,..., x c t j t + + t x c, x 2 c,..., x c. Theorem 3.4. Let x, x 2,..., x be elemets i the -ormed space X. The u x + x x is the oly uique elemet i X satisfyig the followig relatios:. x u, x c, x 2 c,..., x j c, x j+ c,..., x c x c,..., x c for all j {2, 3,..., }, 2. x 2 u, x 2 c, x 3 c,..., x c x c, x 2 c,..., x c, ad 3. x u, x c, x 2 c,..., x c x c, x 2 c,..., x c for some c X with x c, x 2 c,..., x c 0. Proof. Choose t i for all i, 2,..., i Propositio 3.3. The satisfies u x + + x. x u, x c, x 2 c,..., x j c, x j+ c,..., x c x c,..., x c for all j {2, 3,..., }, 2. x 2 u, x 2 c, x 3 c,..., x c x c, x 2 c,..., x c, ad 3. x u, x c, x 2 c,..., x c x c, x 2 c,..., x c. To prove the uiqueess, assume that v is a elemet i X such that x, x 2,..., x, v are -colliear ad v satisfies. x v, x c, x 2 c,..., x j c, x j+ c,..., x c x c,..., x c for all j {2, 3,..., }, 2. x 2 v, x 2 c, x 3 c,..., x c x c, x 2 c,..., x c, ad 3. x v, x c, x 2 c,..., x c x c, x 2 c,..., x c.

6 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), Sice x v, x 2 v,..., x v are liearly depedet, there are scalars λ, λ 2,..., λ such that v λ x + λ 2 x λ x λ + λ λ. Followig the same argumet i the proof of Propositio 3.3 we coclude that v satisfies. x v, x c,..., x j c, x j+ c,..., x c λ j i λ i x c,..., x c, for all j {2, 3,..., }, 2. x 2 v, x 2 c, x 3 c,..., x c λ i λ i x c, x 2 c,..., x c, ad 3. x v, x c, x 2 c,..., x c λ i λ i x c, x 2 c,..., x c. So for ay j, 2,...,, we have Therefore Hece we get Therefore λ j λ + λ λ. λ λ 2 λ λ + + λ. λ λ + λ λ λ + λ λ λ + λ λ }{{} times λ. λ + λ λ λ + λ λ. So we get that λ, λ 2,..., λ are all positive or all egative. I both cases we get that v u. The followig corollary is a direct applicatio to Propositios 3. ad 3.3. Corollary 3.5. Give x,..., x X. Let u t x + t 2 x t x t + t t for some t, t 2,..., t ot all zero. The u satisfies the followig relatios:. t j x c, x 2 c,..., x j c, x j u, x j+ c,..., x t i x u, x c,..., x j c, x j+ c,..., x c, for all j {2, 3,..., x }, 2. t x 2 u, x 2 c,..., x c i2 t i x u, x 2 c,..., x c, 3. t x c, x 2 c,..., x c, x u i t i x u, x c, x 2 c,..., x c for some c X with x c, x 2 c,..., x c 0. Our ext result shows that the Riesz theorem holds whe X is a liear -ormed space.

7 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), Theorem 3.6. Let Z ad W be subspaces of a liear -ormed space X ad W be a -closed proper subset of Z with codimesio greater tha or equal. For each θ (0, ), there are elemets z, z 2,..., z Z such that z, z 2,..., z ad for all w W. z w, z 2 w,..., z w θ Proof. Let v, v 2,..., v Z W be liearly idepedet. Let a if w W v w, v 2 w,..., v w. If a 0, the by defiitio of a -closed set, there is w 0 W such that v w 0, v 2 w 0,..., v w 0 0. Sice v, v 2,..., v are liearly idepedet we get that w 0 0. Sice w 0 W, we have v, v 2,..., w 0 are liearly idepedet. O the other had, sice v w 0, v 2 w 0,..., v w 0 0, we coclude that v w 0, v 2 w 0,..., v w 0 are liearly depedet. Hece v, v 2,..., v, w 0 are liearly depedet which is a cotradictio. So a > 0. Give θ (0, ). Sice a θ > a, there exists w 0 W such that Let For each i {, 2,..., }, let The Also, we have for all w W. a v w 0, v 2 w 0,..., v w 0 < a θ. γ v w 0, v 2 w 0,..., v w 0. z i v i w 0. γ z, z 2,..., z γ v w 0, v 2 w 0,..., v w 0. v w 0 z w, z 2 w,..., z w w,..., v w 0 w γ γ v w 0 γ w,..., v w 0 γ w γ v (w 0 + γ w),..., v (w 0 + γ w) γ > θ v (w 0 + γ w),..., v (w 0 + γ w) a > θ a a θ

8 W. Shataawi, M. Postolache, J. Noliear Sci. Appl. 9 (206), Ope Problems Questio. Is u i Remark 3.2 uique? Questio 2. Let x, x 2,..., x be elemets i the -ormed space X. As a applicatio to Corollary 3.5, satisfies the followig equalities: u x + x x. x c, x 2 c,..., x j c, x j u, x j+ c,..., x ( ) x u, x c,..., x j c, x j+ c,..., x c, for all j {2, 3,..., x }, 2. x 2 u, x 2 c,..., x c ( ) x u, x 2 c,..., x c, 3. x c, x 2 c,..., x c, x u ( ) x u, x c, x 2 c,..., x c for some c X with x c, x 2 c,..., x c 0. Is u uique? Refereces [] A. D. Aleksadrov, Mappigs of families of sets, Soviet Math. Dokl., (970), [2] X. Y. Che, M. M. Sog, Characterizatios o isometries i liear -ormed spaces, Noliear Aal., 72 (2009), [3] H. Y. Chu, O the Mazur-Ulam problem i liear 2-ormed spaces, J. Math. Aal. Appl., 327 (2007), [4] H. Y. Chu, S. K. Choi, D. S. Kag, Mappigs of coservative distaces i liear -ormed spaces, Noliear Aal., 70 (2009), , 2.2 [5] H. Y. Chu, S. H. Ku, D. S. Kag, Characterizatios o 2-isometries, J. Math. Aal. Appl., 340 (2008), , 2 [6] H. Y. Chu, K. H. Lee, C. K. Park, O the Aleksadrov problem i liear -ormed spaces, Noliear Aal., 59 (2004), [7] H. Y. Chu, C. G. Park, W. G. Park, The Aleksadrov problem i liear 2-ormed spaces, J. Math. Aal. Appl., 289 (2004), [8] Y. M. Ma, The Aleksadrov problem for uit distace preservig mappig, Acta Math. Sci. Ser. B Egl. Ed., 20 (2000), [9] B. Mielik, Th. M. Rassias, O the Aleksadrov problem of coservative distaces, Proc. Amer. Math. Soc., 6 (992), 5 8. [0] A. Misiak, -ier product spaces, Math. Nachr., 40 (989), [] A. Misiak, Orthogoality ad orthoormality i -ier product spaces, Math. Nachr., 43 (989), , 2. [2] Th. M. Rassias, O the A. D. Aleksadrov problem of coservative distaces ad the Mazur-Ulam theorem, Noliear Aal., 47 (200), [3] Th. M. Rassias, P. Šemrl, O the Mazur-Ulam problem ad the Aleksadrov problem for uit distace preservig mappigs, Proc. Amer. Math. Soc., 8 (993),