Filoma 26:4 (212), 859 87 DOI 1.2298/FIL124859K Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Carlsson ype orhogonaliy and characerizaion of inner produc spaces Eder Kikiany a,b,c, Sever S. Dragomir a,c a School of Compuaional & Applied Mahemaics, Universiy of he Wiwaersrand, Privae Bag-3, Wis-25, Johannesburg, Souh Africa b Naional ICT Ausralia, The Universiy of Melbourne, Parkville, VIC 31, Ausralia. c School of Engineering and Science, Vicoria Universiy, PO Box 14428, Melbourne Ciy, VIC 81, Ausralia Absrac. In an inner produc space, wo vecors are orhogonal if heir inner produc is zero. In a normed space, numerous noions of orhogonaliy have been inroduced via equivalen proposiions o he usual orhogonaliy, e.g. orhogonal vecors saisfy he Pyhagorean law. In 21, Kikiany and Dragomir [9] inroduced he p-hh-norms (1 p < ) on he Caresian square of a normed space. Some noions of orhogonaliy have been inroduced by uilizing he 2-HH-norm [1]. These noions of orhogonaliy are closely relaed o he classical Pyhagorean orhogonaliy and Isosceles orhogonaliy. In his paper, a Carlsson ype orhogonaliy in erms of he 2-HH-norm is considered, which generalizes he previous definiions. The main properies of his orhogonaliy are sudied and some useful consequences are obained. These consequences include characerizaions of inner produc space. 1. Inroducion In an inner produc space (X,, ), a vecor x X is said o be orhogonal o y X (denoed by x y) if he inner produc x, y is zero. In he general seing of normed spaces, numerous noions of orhogonaliy have been inroduced via equivalen proposiions o he usual orhogonaliy in inner produc spaces, e.g. orhogonal vecors saisfy he Pyhagorean law. For more resuls on oher noions of orhogonaliy, heir main properies, and he implicaions as well as equivalen saemens amongs hem, we refer o he survey papers by Alonso and Beniez [1, 2]. The following are he main properies of orhogonaliy in inner produc spaces (we refer o he works by Alonso and Beniez [1], James [6], and Paringon [12] for references). In he sudy of orhogonaliy in normed spaces, hese properies are invesigaed o see how close he definiion is o he usual orhogonaliy. Suppose ha (X,, ) is an inner produc space and x, y, z X. Then, 1. if x x, hen x = (nondegeneracy); 2. if x y, hen λx λy for all λ R (simplificaion); 3. if (x n ), (y n ) X such ha x n y n for every n N, x n x and y n y, hen x y (coninuiy); 4. if x y, hen λx µy for all λ, µ R (homogeneiy); 5. if x y, hen y x (symmery); 21 Mahemaics Subjec Classificaion. Primary 46B2; Secondary 46C15 Keywords. Carlsson orhogonaliy, characerizaion of inner produc space, Pyhagorean orhogonaliy, Isosceles orhogonaliy Received: 12 February 211; Acceped: 3 Ocober 211 Communicaed by Dragan S. Djordjević Email addresses: eder.kikiany@research.vu.edu.au (Eder Kikiany a,b,c ), sever.dragomir@vu.edu.au (Sever S. Dragomir a,c )
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 86 6. if x y and x z, hen x (y + z) (addiiviy); 7. if x, hen here exiss λ R such ha x (λx + y) (exisence); 8. he above λ is unique (uniqueness). Any pair of vecors in a normed space (X, ) can be viewed as an elemen of he Caresian square X 2. The space X 2 is again a normed space, when i is equipped wih any of he well known p-norms. In 28, Kikiany and Dragomir [9] inroduced he p-hh-norms (1 p < ) on X 2 as follows: (x, y) p HH := ( ) 1 p (1 )x + y p d, for any (x, y) X. These norms are equivalen o he p-norms. However, unlike he p-norms, hey do no depend only on he norms of he wo elemens in he pair, bu also reflec he relaive posiion of he wo elemens wihin he original space X. Some new noions of orhogonaliy have been inroduced by using he 2-HH-norm [1]. These noions of orhogonaliy are closely relaed o he Pyhagorean and Isosceles orhogonaliies (cf. James [6]). The resuls are summarized as follows. Le (X, ) be a normed space. 1. A vecor x X is HH-P-orhogonal o y X (denoed by x HH P y) iff (1 )x + y 2 d = 1 3 (x2 + y 2 ); (1.1) 2. A vecor x X is HH-I-orhogonal o y X (denoed by x HH I y) iff (1 )x + y 2 d = (1 )x y 2 d; (1.2) 3. The homogeneiy (or addiiviy) of he HH-P-(and HH-I-) orhogonaliy characerizes inner produc space. The Pyhagorean and Isosceles orhogonaliies have been generalized by Carlsson in 1962 [4]. In a normed space, x is said o be C-orhogonal o y (denoed by x y (C)) if and only if m α i β i x + γ i y 2 =, where α i, β i, γ i are real numbers such ha m α i β 2 i = m α i γ 2 i = and m α i β i γ i = 1 for some m N. Carlsson s orhogonaliy saisfies he following properies (cf. Alonso and Beniez, [1]; and Carlsson [4]): 1. C-orhogonaliy saisfies nondegeneracy, simplificaion, and coninuiy; 2. C-orhogonaliy is symmeric in some cases (e.g. Pyhagorean and Isosceles orhogonaliies) and no symmeric in oher cases (e.g., x y (C) when x + 2y = x 2y); 3. C-orhogonaliy is eiher homogeneous or addiive o he lef iff he underlying normed space is an inner produc space; 4. C-orhogonaliy is exisen o he righ and o he lef; 5. wih regards o uniqueness, C-orhogonaliy is non-unique when he space is non-round; in paricular, P-orhogonaliy is unique, and I-orhogonaliy is unique iff he underlying normed space is round.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 861 In his paper, we consider a noion of Carlsson s orhogonaliy in HH-sense (which will be called HH-Corhogonaliy), which also generalizes HH-I- and HH-P-orhogonaliies. We discuss is main properies in Secion 3. Some characerizaions of inner produc spaces are provided in Secion 3. Our approach follows ha of Carlsson s [4], considering a condiion which is weaker han homogeneiy and addiiviy of he orhogonaliy. I will be shown ha his condiion implies ha he norm is induced by an inner produc. Consequenly, he homogeneiy (and addiiviy) of his orhogonaliy characerizes inner produc spaces. 2. HH-C-orhogonaliy Moivaed by he relaion beween P-orhogonaliy and HH-P-orhogonaliy (also, hose of I-orhogonaliy and HH-I-orhogonaliy) as saed in Secion 1, we consider a Carlsson ype orhogonaliy in erms of he 2-HH-norm. Le x and y be wo vecors in X and [, 1]. Suppose ha (1 )x y (C), almos everywhere on [, 1], i.e. m α i (1 )β i x + γ i y 2 =, for some m N and real numbers α i, β i, γ i such ha m α i β 2 i = m α i γ 2 i = and m α i β i γ i = 1. (2.1) Then, m α i (1 )β i x + γ i y 2 d =. (2.2) Definiion 2.1. In a normed space (X, ), x X is said o be HH-C-orhogonal o y X (we denoe i by x HH C y) iff x and y saisfy (2.2), wih he condiions (2.1). I can be shown ha he HH-C-orhogonaliy is equivalen o he usual orhogonaliy in any inner produc space. The proof is omied. HH-P-orhogonaliy is a paricular case of HH-C-orhogonaliy, which is obained by choosing m = 3, α 1 = 1, α 2 = α 3 = 1, β 1 = β 2 = 1, β 3 =, γ 1 = γ 3 = 1, and γ 2 =. Similarly, HH-I-orhogonaliy is also a paricular case of HH-C-orhogonaliy, which is obained by choosing m = 2, α 1 = 1 2, α 2 = 1 2, β 1 = β 2 = 1, γ 1 = 1, and γ 2 = 1. We now discuss he main properies of HH-C-orhogonaliy. The following proposiion follows by he definiion of HH-C-orhogonaliy; and we omi he proof. Proposiion 2.2. HH-C-orhogonaliy saisfies he nondegeneracy, simplificaion, and coninuiy. Wih regards o symmery, HH-C-orhogonaliy is symmeric in some cases, for example, HH-P- and HH-I-orhogonaliies are symmeric [1]. The following provides an example of a nonsymmeric HH-Corhogonaliy. Example 2.3. HH-C-orhogonaliy is no symmeric. Proof. Define x HH C2 y o be (1 )x + 2y 2 d = (1 )x 2y 2 d. In R 2 wih l 1 -norm, x = (2, 1) is HH-C 2 -orhogonal o y = ( 1 2, 1) bu y HH C 2 x.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 862 Therefore, i is imporan o disinguish he exisence (as well as addiiviy) o he lef and o he righ. Since HH-P- and HH-I-orhogonaliies are neiher addiive nor homogeneous [1], we conclude ha HH-C-orhogonaliy is neiher addiive nor homogeneous. We will discuss hese properies furher in Secion 3 wih regards o some characerizaions of inner produc spaces. The following lemma is due o Carlsson [4, p. 299]; and i will be used in proving he exisence of HH-C-orhogonaliy. Lemma 2.4. ([4]) Le x, y X. Then, [ λ ± λ 1 (λ + a)x + y 2 λx + y 2] = 2ax 2. Theorem 2.5. Le (X, ) be a normed space. Then, HH-C-orhogonaliy is exisen. Proof. The proof follows a similar idea o ha of Carlsson [4, p. 31]. We only prove for he exisence o he righ, since he oher case follows analogously. Le be a funcion on R defined by (λ) := m α i (1 )β i x + γ i (λx + y) 2 d, where α i, β i, and γ i are real numbers ha saisfy (2.1). Noe ha our domain of inegraion is on (, 1) (we exclude he exremiies) o ensure ha we can employ Lemma 2.4. Therefore, for any λ, m λ 1 (λ) = λ 1 α i (1 )β i x + γ i (λx + y) 2 d (2.3) m [ = λ 1 α i (1 )βi x + γ i (λx + y) 2 γ i (λx + y) 2] d. Noe he use of m α iγ 2 i =. Therefore, (2.3) becomes λ 1 (λ) = λ 1 α i γ i = λ 1 α i γ i [λ + (1 )β i γ 1 i ]γ i x + γ i y 2 λγ i x + γ i y 2 d + [λ + (1 )β i γ 1 i ]γ i x + γ i y 2 λγ i x + γ i y 2 d + 1 3 1 α i (1 )β i x 2 d γ i = α i β 2 i x2. γ i = Noe ha By using Lemma 2.4, we obain λ ± 1 3 λ 1 γ i = α i β 2 i x2 =. λ ± λ 1 (λ) = 2α i γ i (1 )β i γ 1 i γ i x 2 d = 1 3 α i β i γ i x 2 = 1 3 x2, since m α iβ i γ i = 1. I follows ha (λ) is posiive for sufficienly large posiive number λ, and negaive for sufficienly large negaive number λ. By he coninuiy of, we conclude ha here exiss an λ such ha (λ ) =, as required. γ i
3. Characerizaion of inner produc spaces E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 863 The main resul of his secion is a characerizaion of inner produc spaces via he homogeneiy (or addiiviy o he lef) of HH-C-orhogonaliy. Theorem 3.1. Le (X, ) be a normed space in which HH-C-orhogonaliy is homogeneous (or addiive o he lef). Then, X is an inner produc space. Our approach follows ha of Carlsson [4]. The proof of his heorem is described in his secion in wo separae cases: he case for normed spaces of dimension 3 and higher, and he 2-dimensional case. In boh cases, we consider a propery inroduced by Carlsson [4, p. 31], which is weaker han homogeneiy and addiiviy of he orhogonaliy. The following is a modified definiion of he propery. Definiion 3.2. HH-C-orhogonaliy is said o have propery (H) in a normed space X, if x HH C y implies ha n n 1 I is obvious ha m α i nβ i (1 )x + γ i y 2 d =. (3.1) 1. If HH-C-orhogonaliy is homogeneous (or addiive o he lef) in X, hen i has propery (H); 2. If X is an inner produc space, hen HH-C-orhogonaliy is homogeneous (or addiive) and herefore, i has propery (H). Thus, in order o prove Theorem 3.1, i is sufficien o show ha if he HH-C-orhogonaliy has propery (H) in X, hen X is an inner produc space. 3.1. The case of dimension 3 and higher Before saing he proof, recall ha wo vecors x, y in a normed space (X, ) is said o be orhogonal in he sense of Birkhoff (B-orhogonal) if and only if x x + λy for any λ R. Birkhoff s orhogonaliy has a close connecion o he smoohness of he given normed space. Le us recall he definiion of smoohness. In any normed space X, he Gâeaux laeral derivaives of he norm a a poin x X \ {}, i.e. he following is x + y x x + y x τ + (x, y) := and τ + (x, y) := exis for all y X [11, p. 483 485]. The norm is Gâeaux differeniable a x X \ {} if and only if τ + (x, y) = τ (x, y), for all y X. A normed linear space (X, ) is said o be smooh if and only if he norm is Gâeaux differeniable on X \ {}. Lemma 3.3. ([8]) Le (X, ) be a normed space. If he norm is Gâeaux differeniable, hen x is B-orhogonal o y iff τ(x, y) =. In general, B-orhogonaliy is homogeneous. However, i is no always symmeric. In normed spaces of dimension 3 and higher, he symmery of B-orhogonaliy characerize inner produc spaces [1, p. 5]. Therefore, in proving Theorem 3.1, i is sufficien o show ha he propery (H) of HH-C-orhogonaliy implies ha his orhogonaliy is symmeric and equivalen o Birkhoff orhogonaliy in normed spaces of dimension 3 and higher. The following proposiions will be used in proving he heorem (see Lemma 2.6. and Lemma 2.7. of Carlsson [4] for he proof).
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 864 Proposiion 3.4. ([4]) Le (X, ) be a normed space. Recall he following noaion τ ± (x, y) := +( ) x + y x i.e. he righ-(lef-)gâeaux differeniable a x X \ {}. For λµ > we have τ + (λx, µy) = µ τ + (x, y) and τ (λx, µy) = µ τ (x, y); and for λµ < τ + (λx, µy) = µ τ (x, y) and τ (λx, µy) = µ τ + (x, y). Proposiion 3.5. ([4]) If (X, ) is a normed linear space and here exis wo real numbers λ and µ wih λ + µ, such ha λτ + (x, y) + µτ (x, y) is a coninuous funcion of x, y X, hen he norm is Gâeaux differeniable. We will sar wih he following lemma, which also gives us he uniqueness of he HH-C-orhogonaliy. Lemma 3.6. Le (X, ) be a normed space where HH-C-orhogonaliy has propery (H). Suppose ha for any x, y X, here exiss λ R such ha x HH C (λx + y). Then λ = x 1 α i β i γ i τ + (x, y) + α i β i γ i τ (x, y). β i γ i > Proof. By assumpion, we have n n 1 β i γ i < m α i nβ i (1 )x + γ i (λx + y) 2 d =. (3.2) Noe ha by Lemma 2.4, we have he following for any i, and (, 1) (again, noe ha we exclude he exremiies o ensure ha we can employ Lemma 2.4) n 1 [nβ i (1 ) + γ i λ]x + γ i y 2 = n 1 nβ i (1 )x + γ i y 2 + 2β i (1 )γ i λx 2 + ε i (n), (3.3) where ε i (n) when n. Now, we muliply (3.3) by α i and inegrae i over (, 1), o ge n 1 α i [nβ i (1 ) + γ i λ]x + γ i y 2 d (3.4) = n 1 α i nβ i (1 )x + γ i y 2 d + 2α i β i γ i λx 2 (1 ) d + ε i (n). Take he sum and le n o ge = n 1 n m α i nβ i (1 )x + γ i y 2 d + 1 3 λx2 (3.5) (noe he use of (3.4) and m α iβ i γ i = 1). Now, noe ha m α iβ 2 =, and herefore i m n 1 α i nβ i (1 )x + γ i y 2 d m = n 1 α i [nβ i (1 )x + γ i y 2 nβ i (1 )x 2 ]d = m α i [nβ i (1 )x + γ i y nβ i (1 )x]n 1 [nβ i (1 )x + γ i y + nβ i (1 )x]d.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 865 Rewrie nβ i (1 )x + γ i y nβ i (1 )x as n ( β i (1 )x + 1 n γ iy β i (1 )x ), o obain β i (1 )x + 1 n n γ iy β i (1 )x ) = β i (1 )x + sγ i y β i (1 )x s + s = τ + (β i (1 )x, γ i y). Noe also ha Thus, n n 1 [nβ i (1 )x + γ i y + nβ i (1 )x] = [β i (1 )x + n 1 γ i y + β i (1 )x] n n n 1 Therefore, = 2β i (1 )x. m m α i nβ i (1 )x + γ i y 2 d = 2 α i τ + (β i (1 )x, γ i y)β i (1 )xd. m λ = 3x 2 α i τ + (β i (1 )x, γ i y)2β i (1 )x d m = 6x 1 α i β i (1 )τ + (β i (1 )x, γ i y) d. By Proposiion 3.4, (3.5) gives us λ = 6x 1 α i β i γ i τ + (x, y) + α i β i γ i τ (x, y) β i γ i > β i γ i < = x 1 α i β i γ i τ + (x, y) + α i β i γ i τ (x, y), β i γ i > and he proof is compleed. β i γ i < (1 ) d Now, we have a unique λ for any x, y X such ha x HH C λx + y. As a funcion of x and y, λ = λ(x, y) is a coninuous funcion [4, p. 33]. Thus, α i β i γ i τ + (x, y) + α i β i γ i τ (x, y) β i γ i > β i γ i < is also a coninuous funcion in x, y X. By Proposiion 3.5, he norm is Gâeaux differeniable. Togeher wih Lemma 3.3, we have he following consequence. Corollary 3.7. If HH-C-orhogonaliy has propery (H), hen he norm of X is Gâeaux differeniable and x HH C y holds if and only if τ(x, y) =, i.e. x y (B). Remark 3.8. We noe ha he funcion τ(, ) is also coninuous as a funcion of x and y. Le us assume ha x is HH-C-ani-orhogonal o y if and only if y HH C x. We have shown ha when HH-C-orhogonaliy has propery (H), hen i is equivalen o B-orhogonaliy and herefore is homogeneous, since B-orhogonaliy is homogeneous (in any case). This fac implies ha HH-C-ani-orhogonaliy has propery (H) as well. Therefore, he above resuls also hold for HH-C-ani-orhogonaliy. In paricular, τ(x, y) = implies ha x HH C y, i.e., y is HH-C-ani-orhogonal o x; hence τ(y, x) =. Thus, B-orhogonaliy is symmeric, and we obain he following consequence. Corollary 3.9. If HH-C-orhogonaliy has propery (H), hen i is symmeric and equivalen o B-orhogonaliy.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 866 3.2. The 2-dimensional case Previously, we have defined ha x HH C y, when x and y saisfy he following: m α i (1 )β i x + γ i y 2 d =, where m α i β 2 i = m α i γ 2 i = and m α i β i γ i = 1. In his subsecion, we use a slighly differen noaion, in order o resolve he 2-dimensional problem. Noe ha m α i (1 )β i x + γ i y 2 d (3.6) = β i,γ i α i β 2 i (1 )x + γ i y β 1 2 d + 1 3 β i,γ i = Since m α iβ 2 i = m α iγ 2 i =, hen we may rewrie (3.6) as m α i (1 )β i x + γ i y 2 d = β i,γ i α i β 2 i α i β 2 i x2 + 1 3 (1 )x + γ i y β 1 We se p i = α i β 2 i and q i = γ i /β i, and rearrange he indices, = 2 d 1 3 β i =,γ i β i,γ i α i γ 2 i y2. (3.7) α i β 2 i x2 + β i,γ i α i γ 2 i y2. m α i (1 )β i x + γ i y 2 d (3.8) r p k (1 )x + q k y 2 1 d 3 r p k x 2 1 3 r p k q 2 k y2. Assume ha HH-C-orhogonaliy has propery (H). Then, i is equivalen o B-orhogonaliy, and herefore is homogeneous. Denoe S X o be he uni circle of X and le x, y S X such ha x HH C y. Then, (3.8) gives us 3 r p k (1 )x + q k αy 2 d = C1 + C 2 α 2, where C 1 = r p k and C 2 = r p kq 2. We may conclude ha he funcion k ϕ(α) = 3 (1 )x + αy 2 d is he soluion of he funcional equaion r p k F(q k α) = C 1 + C 2 α 2, < α <, (3.9) where r p k = C 1, r p k q 2 k = C 2, r p k q k = 1, q k, k = 1,..., r. (3.1)
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 867 We noe ha he funcion ϕ is coninuously differeniable from Corollary 3.7 and Remark 3.8. In he following resuls by Carlsson [4], i is shown ha he behaviour of ϕ for large and small values of α gives us an explici formula for ϕ. Definiion 3.1. Given a funcional equaion r p k F(q k α) = C 1 + C 2 α 2, < α <, for some real numbers C 1, C 2, p k and q k. We say ha he equaion is symmerical if i can be wrien in he form s s m k F(n k α) m k F( n k α) = C 1 + C 2 α 2 for some real numbers C 1, C 2, m k and n k ; oherwise, i is non-symmerical. Lemma 3.11. ([4]) Le ϕ(α) be a coninuously differeniable soluion of he funcional equaion (3.9) saisfying (3.1) and ϕ(α) = 1 + O(α 2 ) when α ϕ(α) = α 2 + O(α) when α ±. If (3.9) is non-symmerical, hen ϕ(α) = 1 + α 2 for < α <. If (3.9) is (non-rivially) symmerical, hen ϕ(α) = ϕ( α) for < α <. A 2-dimensional normed space has cerain properies ha enable us o work on a smaller subse. One of he useful properies is saed in Lemma 3.12. Before saing he lemma, le us recall ha he norm : X R is said o be Fréche differeniable a x X if and only if here exiss a coninuous linear funcional φ x on X such ha x + z x φ x(z) =. z z I is said o be wice (Fréche) differeniable a x X if and only if here exiss a coninuous bilinear funcional φ x on X 2 such ha x + z x φ x(z) φ x (z, z) =. z z 2 Lemma 3.12. ([3]) If (X, ) is a 2-dimensional normed space, hen he norm is wice differeniable almos everywhere in he uni circle S X = {u X : u = 1}. This resul follows by he fac ha he direcion of he lef-side angen is a monoone funcion, and herefore, by Lebesgue s heorem, is differeniable almos everywhere [3, p. 22]. For us o prove Theorem 3.1 for he 2-dimensional normed spaces, we work on he assumpion ha he normed space has propery (H). Thus, we only need o consider he uni vecors, as he resuls will hold for all vecors due o homogeneiy. Furhermore, he previous proposiion enables us o consider he vecors in a dense subse of he uni circle. The following lemma will be employed in he proof of Theorem 3.1. Lemma 3.13. ([2]) Le (X, ) be a normed space. If an exising orhogonaliy implies Robers orhogonaliy, ha is, x X is R-orhogonal o y X iff x + λy = x λy, for all λ R, hen X is an inner produc space.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 868 Proof of Theorem 3.1 for 2-dimensional case Since HH-C-orhogonaliy has propery (H), i is equivalen o B-orhogonaliy, and herefore is homogeneous. Since dim(x) = 2, he norm is wice differeniable for almos every u S X. Le D be he subse of S X consiss of all poins where he norm is wice differeniable. Le x D and x HH C y (or, equivalenly x y (B)) wih y = 1. Then, he funcion ϕ(α) = 3 (1 )x + αy 2 d is a coninuously differeniable soluion of he funcional equaion (3.9) saisfying (3.1). Claim 3.14. The funcion ϕ saisfies ϕ(α) = 1 + O(α 2 ) when α ϕ(α) = α 2 + O(α) when α ±. The proof of claim will be saed in he end of his secion as Lemmas 3.15 and 3.16. Case 1: Equaion (3.1) is non-symmerical. I follows from Lemma 3.11 ha ϕ(α) = 1 + α 2. If we choose x and y as he uni vecors of a coordinae sysem in he plane X and wrie w = αx + βy, we see ha w = 1 iff α 2 + β 2 = 1. This means ha he uni circle has he equaion α 2 + β 2 = 1, i.e. an Euclidean circle. Therefore, X is an inner produc space. Case 2: Equaion (3.1) is symmerical. I follows from Lemma 3.11 ha ϕ(α) = ϕ( α) for all α, i.e. (1 )x + αy 2 d = (1 )x αy 2 d (3.11) holds for any α R, x, y X where x D and x HH C y. Since D is a dense subse of C and HH-Corhogonaliy is homogeneous, we conclude ha (3.11) also holds for any x X where x HH C y. Le (, 1), and choose α = (1 ) β, hen (3.11) gives us or equivalenly (1 )x + (1 )βy 2 d = (1 )x (1 )βy 2 d, x + βy = x βy, i.e. x y (R). We conclude ha HH-C-orhogonaliy implies R-orhogonaliy. Since HH-C-orhogonaliy is exisen, X is an inner produc space. The proof of claim is saed as he following lemmas: Lemma 3.15. Le (X, ) be a 2-dimensional normed space, and denoe is uni circle by S X. Le u, v S X. Then, he funcion ϕ(α) = 3 (1 )u + αv 2 d saisfies he condiion ϕ(α) = α 2 + O(α) when α ±.
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 869 Proof. For any u, v S X and α R, we have ( ϕ(α) α 2 1 = 3 (1 )u + αv 2 d 1 ) 3 α2 3 (1 )u + αv2 αv 2 d ( ) = 3 (1 )u + αv αv (1 )u + αv + αv d 3 = 3 (1 ) u Thus, ϕ(α) α 2 = O(α), when α ±. ( ) (1 )u + 2αv d ((1 ) 2 + 2(1 ) α ) d = 1 + α. Lemma 3.16. Le (X, ) be a 2-dimensional normed space and denoe is uni circle by S X. Then, here is a dense subse D of S X such ha if u D and u v (B), he funcion saisfies ϕ(α) = 3 (1 )u + αv 2 d ϕ(α) = 1 + O(α 2 ) when α. (3.12) Proof. Since dim(x) = 2, hen he norm is wice differeniable for almos every u S X by Lemma 3.12 [3, p. 22]. Le D be he subse of S X consiss of all poins where he norm is wice differeniable. We conclude ha D is a dense subse of S X. Denoe φ(x) = x, hen for any u D, he derivaive φ u is a linear funcional and he second derivaive φ u is a bilinear funcional. Furhermore, we have he following u + z u φ u(z) φ u (z, z) z z 2 =. (3.13) Le u D and u v (B), where v = 1. Se z = u v (B), φ u(v) =, and (3.13) gives us u + ( 1 α) v u ( 1 α) 2 φ u (v, v) α ( 1 α) 2 =, v 2 i.e., for any ϵ >, here exiss δ >, such ha for any α < δ u + ( 1 α) v 1 ( φ 1 α) 2 u (v, v) < ϵ. Furhermore, u + (1 ) αv 1 2 α (1 ) 2 2 < ϵ + φ u (v, v) = M. 1 αv ( (, 1)), herefore, when α, z. Since
E. Kikiany, S.S. Dragomir / Filoma 26:4 (212), 859 87 87 Equivalenly, we have, u + 1 αv 1 < M 2 (1 ) 2 α2. Noe ha for any (, 1), u + 1 αv + 1 2 when α. Se ϵ = 1, hen here exiss δ1 such ha for any α < δ 1, we have u + 1 αv + 1 2 < 1, i.e., u + 1 αv + 1 < 1 + 2 = 3. Now, for any α < min{δ, δ 1 }, we have ( ϕ(α) 1 = 3 (1 )u + αv 2 d 1 3) 3 (1 )u + αv2 (1 )u 2 d = 3 (1 ) 2 u + 1 αv 2 u 2 d ( = 3 (1 ) 2 u + 1 αv ) + 1 u + 1 αv 1 d < 9 i.e., ϕ(α) 1 = O(α 2 ), when α. (1 ) 2 M 2 (1 ) 2 α2 d = 3Mα 2, The las wo resuls conclude ha he homogeneiy (also, he righ-addiiviy) of HH-C-orhogonaliy is a necessary and sufficien condiion for he normed space o be an inner produc space. References [1] J. Alonso, C. Beníez, Orhogonaliy in normed linear spaces: a survey. I. Main properies, Exraca Mah. 3(1) (1988) 1 15. [2] J. Alonso, C. Beníez, Orhogonaliy in normed linear spaces: a survey. II. Relaions beween main orhogonaliies, Exraca Mah. 4(3) (1989) 121 131. [3] D. Amir, Characerizaions of inner produc spaces, Operaor Theory: Advances and Applicaions, vol. 2, Birkhäuser Verlag, Basel, 1986. [4] S.O. Carlsson, Orhogonaliy in normed linear spaces, Ark. Ma. 4 (1962), 297 318 (1962). [5] S.S. Dragomir, Semi-inner Producs and Applicaions, Nova Science Publishers, Inc., Hauppauge, NY, 24. [6] R.C. James, Orhogonaliy in normed linear spaces, Duke Mah. J. 12 (1945) 291 32. [7] R.C. James, Inner produc in normed linear spaces, Bull. Amer. Mah. Soc. 53 (1947) 559 566. [8] R.C. James, Orhogonaliy and linear funcionals in normed linear spaces, Trans. Amer. Mah. Soc. 61 (1947) 265 292. [9] E. Kikiany, S.S. Dragomir, Hermie-Hadamard s inequaliy and he p-hh-norm on he Caresian produc of wo copies of a normed space, Mah. Inequal. Appl. 13(1) (21) 1 32. [1] E. Kikiany, S.S. Dragomir, Orhogonaliy conneced wih inegral means and characerizaions of inner produc spaces, J. Geom. 98(1 2) (21) 33 49. [11] R.E. Megginson, An Inroducion o Banach Space Theory, Graduae Texs in Mahemaics, vol. 183, Springer-Verlag, New York, 1998. [12] J.R. Paringon, Orhogonaliy in normed spaces, Bull. Ausral. Mah. Soc. 33(3) (1986) 449 455.