Some Properties in Generalized n-inner Product Spaces

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 2010, no. 45, Some Properties in Generalized n-inner Product Spaces B. Surender Reddy Department of Mathematics, PGCS, Saifabad Osmania University, Hyderabad , A.P., India Abstract The main aim of this paper is to prove parallelogram law, Polarization identity in generalized n-inner product spaces over the field K = R of real numbers or the field K = C of complex numbers. Mathematics Subject Classification: 46C05, 26D20 Keywords: Cauchy-Schwarz inequality, n-inner product space, n-norm, generalized n-inner product space 1 Introduction The concept of n-inner product and n-inner product spaces has been investigated by A.Misiak [1, 2] and has been developed extensively in different subjects by others, for example [3, 4, 5]. A.Misiak [1] has introduced an n-normed space by the following definition. Definition 1.1. Let n N and let X be a linear space of dimension n over the field K = R of real numbers or the field K = C of complex numbers and,,..., is a real valued function on X } X X {{} = X n such that n nn 1 : x 1,x 2,...,x n =0if and only if x 1,x 2,...,x n are linearly dependent vectors, nn 2 : x 1,x 2,...,x n is invariant under any permutation of x 1,x 2,...,x n, nn 3 : x 1,x 2,...,x n 1,αx n = α x 1,x 2,...,x n for all α K, nn 4 : x 1,x 2,...,x n 1,y+z x 1,x 2,...,x n 1,y + x 1,x 2,...,x n 1,z for all x 1,x 2,...,x n 1,y,z X, then the function,,..., is called an n-norm on X and the pair (X,,,..., ) is called linear n-normed space.

2 2230 B.Surender Reddy A.Misiak [1] has introduced an n-inner product space by the following definition. Definition 1.2. Let n N and X be a linear space of dimension n over the field K = R of real numbers or the field K = C of complex numbers and (, /,,..., ) is a K-valued function defined on X } X X {{} = X n+1 n+1 (n 2) such that ni 1 :(x 1,x 1 /x 2,x 3,...,x n ) 0 for any x 1,x 2,...,x n X and (x 1,x 1 /x 2,x 3,...,x n )=0if and only if x 1,x 2,...,x n are linearly dependent vectors, ni 2 :(a, b/x 1,x 2,...,x n 1 )=(ϕ(a),ϕ(b)/π(x 1 ),π(x 2 ),...,π(x n 1 )) for any a, b, x 1,x 2,...,x n 1 X and for any bijection π : {x 1,x 2,...,x n 1 } {x 1,x 2,...,x n 1 } and ϕ : {a, b} {a, b}, ni 3 : If n>1 then (x 1,x 1 /x 2,x 3,...,x n )=(x 2,x 2 /x 1,x 3,...,x n ) for any x 1,x 2,x 3,...,x n X, ni 4 :(αa, b/x 1,x 2,...,x n 1 )=α(a, b/x 1,x 2,...,x n 1 ) for any a, b, x 1,x 2,...,x n 1 X and for any scalar α K, ni 5 :(a+a 1, b/x 1,x 2,...,x n 1 )=(a, b/x 1,x 2,...,x n 1 )+(a 1, b/x 1,x 2,...,x n 1 ) for any a, a 1,b,x 1,x 2,...,x n 1 X, then (, /,,..., ) is called the n-inner product and (X, (, /,,..., )) is called the n-inner product space. This n-inner product induces an n-norm by x 1,x 2,...,x n = (x 1,x 1 /x 2,...,x n 1 ). If n = 1 then the definition 1.1 reduces to the ordinary inner product. 2 Preliminaries K.Trenceveski and R.Malceski [5] has introduced the following definition of generalized n-inner product space. Definition 2.1. Let n N and X be a linear space of dimension n over the field K = R of real numbers or the field K = C of complex numbers and,,..., /,,..., is a K-valued function defined on X } X X {{} = 2n X 2n such that gni 1 : a 1,a 2,...,a n /a 1,a 2,...,a n > 0 if a 1,a 2,...,a n are linearly independent vectors, gni 2 : a 1,a 2,...,a n /b 1,b 2,...,b n = b 1,b 2,...,b n /a 1,a 2,...,a n for any a 1,a 2,...,a n,b 1,b 2,...,b n X, gni 3 : λa 1,a 2,...,a n /b 1,b 2,...,b n = λ a 1,a 2,...,a n /b 1,b 2,...,b n for any scalar λ K and any a 1,a 2,...,a n,b 1,b 2,...,b n X,

3 Some properties in generalized n-inner product spaces 2231 gni 4 : a 1,a 2,...,a n /b 1,b 2,...,b n = a σ(1),a σ(2),...,a σ(n) /b 1,b 2,...,b n for any odd permutation σ in the set {1, 2,...,n} and any a 1,a 2,...,a n,b 1,b 2,...,b n X, gni 5 : a 1 + c, a 2,...,a n /b 1,b 2,...,b n = a 1,a 2,...,a n /b 1,b 2,...,b n + c, a 2,...,a n /b 1,b 2,...,b n for any a 1,a 2,...,a n,b 1,b 2,...,b n,c X, gni 6 : If a 1,b 1,...,b i 1,b i+1,...,b n /b 1,b 2,...,b n =0for each i {1, 2,...,n} then a 1,a 2,...,a n /b 1,b 2,...,b n =0for arbitrary vectors a 1,a 2,...,a n, then the function,,..., /,,..., is called the generalized n-inner product and the pair (X,,,..., /,,..., ) is called the generalized n-inner product space ( or generalized n-pre Hilbert space). In the special case, if we consider only such pair of sets a 1,a 2,...,a n and b 1,b 2,...,b n which differ for at most one vector, for example a 1 = a, b 1 = b and a 2 = b 2 = x 1, a 3 = b 3 = x 2,..., a n = b n = x n 1 then by putting (a, b/x 1,x 2,...,x n 1 ) = a, x 1,x 2,...,x n 1 /b, x 1,x 2,...,x n 1 we obtain an n-inner product. Example 2.2. Let X be a linear space with inner product / then a 1,b 1 a 1,b 2... a 1,b n a 2,b 1 a 2,b 2... a 2,b n a 1,a 2,...,a n /b 1,b 2,...,b n = a n,b 1 a n,b 2... a n,b n defined the generalized n-inner product on X. K.Trenceveski and R.Malceski [5] proved Cauchy-Schwarz inequality in generalized n-inner product on X as a 1,a 2,...,a n /b 1,b 2,...,b n 2 a 1,a 2,...,a n /a 1,a 2,...,a n b 1,b 2,...,b n /b 1,b 2,...,b n i.e., a 1,a 2,...,a n /b 1,b 2,...,b n a 1,a 2,...,a n b 1,b 2,...,b n The generalized n-inner product on X induces an n-norm by x 1,x 2,...,x n = x 1,x 2,...,x n /x 1,x 2,...,x n and it is the same n-norm induced by the definition 1.2. Let (X,,,..., /,,..., ) be the generalized n-inner product space and,,..., be the induced n- norm. We observe that if x k x then x k,z 2,...,z n x, z 2,...,z n for every z 2,...,z n X. This result tells us that n-norm,,..., is continuous in first variable and by property nn 2 of n-norms,,..., is continuous in each variable.

4 2232 B.Surender Reddy 3 Continuity of generalized n-inner product If x 1k y 1, x 2k y 2,..., x nk y n then by property gni 5 and Cauchy-Schwarz inequality for the generalized n-inner product, we have x 1k,x 2k,...,x nk /z 1,z 2,...,z n y 1,y 2,...,y n /z 1,z 2,...,z n = x 1k,x 2k,...,x nk /z 1,z 2,...,z n y 1,x 2k,...,x nk /z 1,z 2,...,z n + y 1,x 2k,...,x nk /z 1,z 2,...,z n y 1,y 2,x 3k,...,x nk /z 1,z 2,...,z n + y 1,y 2,x 3k,...,x nk /z 1,z 2,...,z n... y 1,y 2,y 3,...,y n /z 1,z 2,...,z n + y 1,y 2,y 3,...,y n /z 1,z 2,...,z n y 1,y 2,y 3,...,y n /z 1,z 2,...,z n = x 1k y 1,x 2k,...,x nk /z 1,z 2,...,z n + y 1,x 2k y 2,x 3k,...,x nk /z 1,z 2,...,z n + + y 1,y 2,...,x nk y n /z 1,z 2,...,z n x 1k y 1,x 2k,...,x nk /z 1,z 2,...,z n + y 1,x 2k y 2,x 3k,...,x nk /z 1,z 2,...,z n y 1,y 2,...,x nk y n /z 1,z 2,...,z n x 1k y 1,x 2k,...,x nk z 1,z 2,...,z n + y 1,x 2k y 2,x 3k,...,x nk z 1,z 2,...,z n y 1,y 2,...,x nk y n z 1,z 2,...,z n 0ask Since we know that n-norms,,..., is continuous in each variable. Hence x 1k,x 2k,...,x nk /z 1,z 2,...,z n y 1,y 2,...,y n /z 1,z 2,...,z n. This shows that,,..., /,,..., is continuous in first n-variable and hence by the property gni 2 and gni 4 we get,,..., /,,..., is continuous in each variable. Now, we give the parallelogram law and Polarization identity in generalized n-inner product space. Proposition 3.1. Parallelogram law in generalized n-inner product space x+y, x 2,...,x n 2 + x y, x 2,...,x n 2 =2 x, x 2,...,x n 2 +2 y, x 2,...,x n 2 Proof. x + y, x 2,...,x n 2 + x y, x 2,...,x n 2 = x + y, x 2,...,x n /x + y, x 2,...,x n + x y, x 2,...,x n /x y, x 2,...,x n = x, x 2,...,x n /x + y, x 2,...,x n + y, x 2,...,x n /x + y, x 2,...,x n + x, x 2,...,x n /x y, x 2,...,x n y, x 2,...,x n /x y, x 2,...,x n = x + y, x 2,...,x n /x, x 2,...,x n + x + y, x 2,...,x n /y, x 2,...,x n + x y, x 2,...,x n /x, x 2,...,x n x y, x 2,...,x n /y, x 2,...,x n

5 Some properties in generalized n-inner product spaces 2233 = x, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /x, x 2,...,x n + x, x 2,...,x n /y, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n + x, x 2,...,x n /x, x 2,...,x n y, x 2,...,x n /x, x 2,...,x n x, x 2,...,x n /y, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n = x, x 2,...,x n /x, x 2,...,x n + x, x 2,...,x n /y, x 2,...,x n + y, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n + x, x 2,...,x n /x, x 2,...,x n x, x 2,...,x n /y, x 2,...,x n y, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n = 2 x, x 2,...,x n /x, x 2,...,x n + 2 y, x 2,...,x n /y, x 2,...,x n = 2 x, x 2,...,x n y, x 2,...,x n 2 Proposition 3.2. Polarization Identity in the generalized n-inner product space 4 x, x 2,...,x n /y, x 2,...,x n = x + y, x 2,...,x n 2 x y, x 2,...,x n 2 = + i x + iy, x 2,...,x n 2 i x iy, x 2,...,x n 2 4 i 4 x + i k y, x 2,x 3,...,x n (where i 2 = 1) k=1 Proof. x + y, x 2,...,x n 2 = x + y, x 2,...,x n /x + y, x 2,...,x n = x, x 2,...,x n /x + y, x 2,...,x n + y, x 2,...,x n /x + y, x 2,...,x n = x + y, x 2,...,x n /x, x 2,...,x n + x + y, x 2,...,x n /y, x 2,...,x n = x, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /x, x 2,...,x n + x, x 2,...,x n /y, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n x+y, x 2,...,x n 2 = x, x 2,...,x n /x, x 2,...,x n + x, x 2,...,x n /y, x 2,...,x n + y, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n (3.1) Similarly, we have x y, x 2,...,x n 2 = x, x 2,...,x n /x, x 2,...,x n x, x 2,...,x n /y, x 2,...,x n y, x 2,...,x n /x, x 2,...,x n + y, x 2,...,x n /y, x 2,...,x n (3.2) On using (3.1) and (3.2), we get x+y, x 2,...,x n 2 x y,x 2,...,x n 2 =2 x, x 2,...,x n /y, x 2,...,x n

6 2234 B.Surender Reddy + 2 y, x 2,...,x n /x, x 2,...,x n (3.3) Put y = iy in equation (3.3) on both sides, we get x + iy, x 2,...,x n 2 x iy, x 2,...,x n 2 = 2 x, x 2,...,x n /iy, x 2,...,x n + 2 iy, x 2,...,x n /x, x 2,...,x n = 2 iy, x 2,...,x n /x, x 2,...,x n + 2i y, x 2,...,x n /x, x 2,...,x n = 2 i y, x 2,...,x n /x, x 2,...,x n + 2i y, x 2,...,x n /x, x 2,...,x n = 2i x, x 2,...,x n /y, x 2,...,x n +2i y, x 2,...,x n /x, x 2,...,x n multiplying the above equation by i on both sides, we get i x + iy, x 2,...,x n 2 i x iy, x 2,...,x n 2 = 2i 2 x, x 2,...,x n /y, x 2,...,x n + 2i 2 y, x 2,...,x n /x, x 2,...,x n = 2 x, x 2,...,x n /y, x 2,...,x n 2 y, x 2,...,x n /x, x 2,...,x n (3.4) adding (3.3) and (3.4) we get x + y, x 2,...,x n 2 x y, x 2,...,x n 2 + i x + iy, x 2,...,x n 2 i x iy, x 2,...,x n 2 = 4 x, x 2,...,x n /y, x 2,...,x n 4 = 4 x, x 2,...,x n /y, x 2,...,x n = i 4 x + i k y, x 2,x 3,...,x n Hence proved. ACKNOWLEDGEMENTS. The author is grateful to the referee for careful reading of the article and suggested improvements. References [1] A. Misiak, n-inner product spaces, Math. Nachr., 140 (1989), [2] A.Misiak, Orthogonality and Orthonormality in n-inner product spaces, Math. Nachr., 143 (1989), [3] H. Gunawan and Mashadi, On n-normed spaces, International J. Math. Math. Sci., 27 (2001), no. 10, [4] H.Gunawan, On n-inner products, n-norms and Cauchy-Schwarz inequality, Sci. Math. Jpn., 5 (2002), [5] K.Trencevski and R.Malceski, On generalized n-inner product and the corresponding Cauchy-Schwarz inequality, Journal of Inequalities in Pure and Applied Math., 7 (2006), Received: August 8, 2010 k=1