In this paper, some necessary and sufficient conditions for an n normed spaces to be an n inner product spaces are given

Al-sayaad, Antar Ali

Hardan, Basel Mohammed

English

Scitech Research Journals

Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218  Volume 1, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm/index                                 36|  SCITECH                                                                     Volume 1, Issue 1  RESEARCH ORGANISATION|                                   January, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com    On  Characterizations  of  𝐧–Inner Product Spaces  Antar Ali Al-sayaad1  and  Basel Mohammed  Hardan2 D1 epartment of Mathematics, Faculty Education of Aden, University of Aden, Yemen. al-sayaad@yandex.com D2 epartment of Mathematics, Faculty Education of Zingebar, University of Aden, Yemen.         bassilhardan@yahoo.com  Abstract In this paper, some necessary and sufficient conditions for an  𝑛– normed spaces to be an  𝑛–inner product spaces are given.  Keywords:  𝑛–inner product spaces; characterizations. 1.  Introduction  Chen  and  Song [1], showed that  a necessary and sufficient conditions  for an  𝑛–normed spaces to be an  𝑛–inner product spaces is the following extended parallelogram law:  𝑥 + 𝑦, 𝑎2 , … , 𝑎𝑛 2 +  𝑥 − 𝑦, 𝑎2 , … , 𝑎𝑛 2 = 2  𝑥, 𝑎2 , … , 𝑎𝑛 2 +  𝑦, 𝑎2 , … , 𝑎𝑛 2                 (1.1)        holds, in such case the  𝑛–inner product spaces is given by    𝑥, 𝑦 𝑎2, … , 𝑎𝑛 =  14  𝑥 + 𝑦, 𝑎2, … , 𝑎𝑛 −  𝑥 − 𝑦, 𝑎2 , … , 𝑎𝑛  .                                       (1.2)  for all  𝑥, 𝑦, 𝑎2, … , 𝑎𝑛 ∈ 𝑋. Recently, Soenjaya [3] called that above law a characterization  of 𝑛–inner product spaces.    For this work we need the following definitions: Definition 2.1.[2] Let 𝑋 be a real vector spaces of   dim ≥ 𝑛.  An 𝑛–norm on 𝑋 is a mapping  ∙, … ,∙ ∶  𝑋𝑛 → ℝ, which satisfies the following four conditions:  𝑛N 1:   𝑥1 , … , 𝑥𝑛 = 0,   if and only if  𝑥1 , … , 𝑥𝑛   are linearly dependent,  𝑛N 2:   𝑥1 , … , 𝑥𝑛 =  𝑥𝑖1 , … , 𝑥𝑖𝑛  ,  for every permutation  𝑖1 , … 𝑖𝑛   of  (1, … , 𝑛),                    𝑛N 3:   𝛼𝑥1 , … , 𝑥𝑛 =  𝛼  𝑥1 , … , 𝑥𝑛   for  𝛼 ∈ ℝ,                              𝑛N 4:   𝑥1 + 𝑥 1 , 𝑥2 , … , 𝑥𝑛 ≤  𝑥1 , 𝑥2, … , 𝑥𝑛 +  𝑥 1 , 𝑥2 , … , 𝑥𝑛 ,        for all  𝑥1 , 𝑥 1 , 𝑥2, … , 𝑥𝑛 ∈ 𝑋.  The pair   𝑋,  ∙,… ,∙   is called  an  𝑛–normed spaces. Definition 2.2. [2]  A real-valued function   ∙,∙  ∙,… ,∙  on  𝑋𝑛+1 satisfied the following properties: Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218  Volume 1, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm/index                                 37| 𝑛I 1:   𝑥1 , 𝑥1 𝑥2 , … , 𝑥𝑛   ≥ 0  and   𝑥1 , 𝑥1 𝑥2 , … , 𝑥𝑛  = 0,                                                                          if and only if   𝑥1 , 𝑥2 … , 𝑥𝑛   are linearly dependent.    𝑛I 2:   𝑥1 , 𝑥1 𝑥2 , … , 𝑥𝑛  =  𝑥𝑖1 , 𝑥𝑖1  𝑥𝑖2 , … , 𝑥𝑖𝑛  , for any permutation   𝑖1, … , 𝑖𝑛  of   1, … , 𝑛 .  𝑛I 3:   𝑥1 , 𝑥1 𝑥2 , … , 𝑥𝑛  =   𝑥1 , 𝑥1  𝑥2, … , 𝑥𝑛  ,                                       𝑛I 4:   𝛼𝑥1 , 𝑥1 𝑥2 , … , 𝑥𝑛  = 𝛼 𝑥1 , 𝑥1 𝑥2, … , 𝑥𝑛  ,   for every 𝛼 ∈ ℝ.    𝑛I 5:   𝑥0 + 𝑥 0 , 𝑥1 𝑥2 , … , 𝑥𝑛  =  𝑥0 , 𝑥1 𝑥2 , … , 𝑥𝑛  +  𝑥 0, 𝑥1 𝑥2, … , 𝑥𝑛  .   is called an  𝑛–inner product  on a vector spaces  𝑋. The pair   𝑋,  ∙,∙  ∙,… ,∙    is called an  𝑛–inner product spaces.    3. Main results   Theorem 3.1. A characterization of  𝑛– inner product spaces  𝑋,  ∙,∙  ∙, … ,∙    where  𝑋,  ∙, … ,∙   be an  𝑛–normed spaces   on  ℂ are: i. 2  𝑥, 𝑥2, … , 𝑥𝑛 2 −  𝑦, 𝑥2 , … , 𝑥𝑛 2 =  𝑥 + 𝑖𝑦, 𝑥2, … , 𝑥𝑛 2 +  𝑥 − 𝑖𝑦, 𝑥2 , … , 𝑥𝑛 2,       (3.1) ii.  𝑥, 𝑦 𝑥2 , … ,𝑥𝑛     =18  𝑥 + 𝑦, 𝑥2 , … , 𝑥𝑛 2 −  𝑥 − 𝑦, 𝑥2 , … , 𝑥𝑛 2 + 𝑖  𝑥 + 𝑖𝑦, 𝑥2 , … , 𝑥𝑛 2 −  𝑥 − 𝑖𝑦, 𝑥2 , … , 𝑥𝑛 2  ,                           (3.2)   for every 𝑥, 𝑦, 𝑥2 , … , 𝑥𝑛 ∈ 𝑋. Proof.     To prove equation (3.1) of (i),  R.H.S. =  𝑥 + 𝑖𝑦, 𝑥2, … , 𝑥𝑛 2 +  𝑥 − 𝑖𝑦, 𝑥2 , … , 𝑥𝑛 2                 =  𝑥 + 𝑖𝑦, 𝑥 + 𝑖𝑦 𝑥2 , … , 𝑥𝑛  +  𝑥 − 𝑖𝑦, 𝑥 − 𝑖𝑦 𝑥2 , … , 𝑥𝑛         =  𝑥, 𝑥 𝑥2, … , 𝑥𝑛  − 2𝑖 𝑥, 𝑦 𝑥2 , … , 𝑥𝑛  −  𝑦, 𝑦 𝑥2 , … , 𝑥𝑛                                                 +             𝑥, 𝑥 𝑥2, … , 𝑥𝑛  + 2𝑖 𝑥, 𝑦 𝑥2 , … , 𝑥𝑛 −  𝑦, 𝑦 𝑥2 , … , 𝑥𝑛            = 2 𝑥, 𝑥 𝑥2, … , 𝑥𝑛  − 2 𝑦, 𝑦 𝑥2, … , 𝑥𝑛                                                        = 2( 𝑥, 𝑥2, … , 𝑥𝑛 2 −  𝑦, 𝑥2, … , 𝑥𝑛 2).                                                   Which is the  L.H.S. of equation (3.1). To prove equation (3.2) of (ii),  R.H.S.   =18  𝑥 + 𝑦, 𝑥2, … , 𝑥𝑛 2 −  𝑥 − 𝑦, 𝑥2, … , 𝑥𝑛 2+ 𝑖  𝑥 + 𝑖𝑦, 𝑥2 , … , 𝑥𝑛 2 −  𝑥 − 𝑖𝑦, 𝑥2, … , 𝑥𝑛 2                     Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218  Volume 1, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm/index                                 38|                      =18 4 𝑥, 𝑦 𝑥2 , … , 𝑥𝑛 + 𝑖 −4𝑖 𝑥, 𝑦 𝑥2 , … , 𝑥𝑛                                        =  𝑥, 𝑦 𝑥2 , … , 𝑥𝑛  ,                                                                                        which is the  L.H.S. of equation (3.2).       References [1]  Chen, X. Y. and Song, M. M.,  Characterizations on isometrics in linear  n-normed, Nonlinear Anal.TMA.72(2010),1895-1901.  [2] Gunawan, H., on  n-inner products, n-norms, and the Cauchy Schwarz inequality., Sci. Math. Jpn. 55(2002),53-60.  [3]  Soenja, A. L., Characterizations of n-inner product spaces, ijpm. 78(7)(2012), 1011-1018. 

On Characterizations of n-Inner Product Spaces

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On Characterizations of n-Inner Product Spaces

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