Some Results in Generalized n-inner Product Spaces

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1 International Mathematical Forum, 4, 2009, no. 21, Some Results in Generalized n-inner Product Spaces Renu Chugh and Sushma 1 Department of Mathematics M.D. University, Rohtak , India Abstract The aim of this paper is to prove parallelogram law, polarization identity in Generalized n-inner product spaces defined by K. Trencevski and R. Malceski [5] which is generalization of n-inner product spaces introduced by A. Misiak [1]. Also we discuss the notion of strong and weak convergence in Generalized n-inner product spaces and study the relation between them. Mathematics Subject Classification: 46C05, 26D20 Keywords: Cauchy-Schwarz inequality, n-inner product, n-norm, generalizedinner product, strong and weak convergence A. Misiak [1] has introduced an n-normed linear space by the following definition Definition 1.1. Let n N (natural numbers) and X be a real linear space of dimension greater than or equal to n. A real valued function,..., on X X... X = X n satisfying the following four properties: (1.1) x 1,x 2,...,x n = 0 if any only if x 1,x 2,...,x n are linearly dependent, (1.2) x 1,x 2,...,x n is invariant under any permutation, (1.3) x 1,x 2,...,ax n = a x 1,x 2,...,x n, for any a R (real), (1.4) 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 is called an n-norm on X and the pair (X,,..., ) is called n-normed linear space. Although in this paper we only consider real vector spaces, the results of this paper can easily be generalized for complex vector spaces. A. Misiak [1] introduced an n-inner product in the following form: 1 lathersushma@yahoo.com

2 1014 R. Chugh and Sushma Definition 1.2. Assume that n is a positive integer and V is a real vector space such that dim X n and (,,..., ) is a real function defined on }{{}} X X {{... X } n 1 n+1 such that: 1. (x 1,x 1 x 2,...,x n ) 0, for any x 1,x 2,...,x n X and (x 1,x 1 x 2,...,x n )=0 if and only if x 1,x 2,...,x n are linearly dependent vectors; 2. (a, b x 1,...,x n 1 ) = (ϕ(a),ϕ(b) π(x 1 ),...,(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}; 3. If n > 1, then (x 1,x 1 x 2,...,x n ) = (x 2,x 2 x 1,x 3,...,x n ), for any x 1,x 2,...,x n X ; 4. (αa, b x 1,...,x n 1 )=α(a, b x 1,...,x n 1 ), for any a, b, x 1,...,x n 1 X and any scalar α R; 5. (a + a 1,b x 1,...,x n 1 ) = (a, b x 1,...,x n 1 )+(a 1,b x 1,...,x n 1 ), for any a, b, a 1,x 1,x 2,...,x n X; Then (,,..., ) is called the n-inner product and (X, (,,..., ) is called }{{}}{{} n 1 n 1 the n-inner product space. This n-inner product induces an n-norm by x 1,...,x n = (x 1,x 1 x 2,...,x n ). K. Trenceveski and R. Malceski [5] has introduced following definition of generalized n-inner product as: Definition 1.3. Assume that n is a positive integer, X is a real vector space such that dim X n and,...,,..., is a real function on X 2n such that (3.1) a 1,...,a n a 1,...,a n > 0ifa 1,...,a n are linearly independent vectors, (3.2) a 1,...,a n b 1,...,b n = b 1,...,b n a 1,...,a n for any a 1,...,a n,b 1,...,b n X (3.3) λa 1,...,a n b 1,...,b n = λ a 1,...,a n b 1,...,b n for any scalar λ R and any a 1,...,a n,b 1,...,b n X, (3.4) a 1,...,a n b 1,...,b n = a σ(1),...,a σ(n) b 1,...,b n for any odd permutation σ in the set {1,...,n} and any a 1,...,a n,b 1,...,b n X

3 Results in generalized n-inner product spaces 1015 (3.5) a 1 + c, a 2,...,a n b 1,...,b n = a 1,a 2...,a n b 1,...,b n + c, a 2,...,a n b 1,...,b n for any a 1,...,a n,b 1,...,b n,c X. (3.6) If a 1,b 1,...,b i 1,b i+1,...,b n b 1,...,b n = 0 for each i {1, 2,...,n}, then a 1,...,a n b 1,...,b n = 0 for arbitrary vectors a 2,...,a n. Then the function,...,,..., is called an generalized n-inner product and the pair (X,,...,,..., ) is called an generalized n-inner product space. In the special case if we consider only such pairs of sets a 1,...,a n and b 1,...,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 n = b n = x n 1, then by putting (a, b x 1,...,x n 1 )=(a, x 1,...,x n 1 b, x 1,...,x n 1 ). we obtain an n-inner product. Example 1.4 ([5]). Let X be a 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 n b 1,...,b n =.... a n b 1 a n b 2... a n b n defines an 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 n b 1,...,b n 2 a 1,...,a n a 1,...a n b 1,...,b n b 1,...,b n The generalized n-inner product on X induces an n-norm by x 1,...,x n = x 1,...x n x 1,...,x n and it is the same n-norm induced by Definition 2.1. Let (X,,...,,..., ) be 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 tells us that n- norm,..., is continuous in first variable.and by property (1.2) of n-norms,..., is continuous in each variable.

4 1016 R. Chugh and Sushma 2 CONTINUITY OF GENERALIZED n-inner PRODUCT Next if x 1k y 1,x 2k y 2,...,x nk y n, then by property (3.5) and Cauchy- Schwarz inequality for 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 nk z 1,z 2,...,z n + y 1,y 2,...,x nk z 1,z 2,...,z n... y 1,y 2,...,y n z 1,z 2,...,z n + y 1,y 2,...,y n z 1,z 2,...,z n y 1,y 2,...,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 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 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 nk z 1,z 2,...,z n y 1,y 2,...,x nk y n z 1,z 2,...,z n 0 as k 0 Since we know that n-norm,..., 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 (3.2) and (3.4) we get,,...,,,..., is continuous in each variable. Now, we give the Parallelogram law and Polarization identity in generalized n-inner product space. Proposition 2.1 (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

5 Results in generalized n-inner product spaces 1017 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, 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 2 +2 y, x 2,...,x n 2. Hence, 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 Proposition 2.2 (Polarization Identity in 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 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, 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 y, x 2,...,x n +2 y, x 2,...,x n x, x 2,...,x n = 2 x, x 2,...,x n y, x 2,...,x n +2 x, x 2,...,x n y, x 2,...,x n = 4 x, x 2,...,x n y, x 2,...,x n

6 1018 R. Chugh and Sushma 3 STRONG AND WEAK CONVERGENCE IN GENERALIZED n-inner PRODUCT SPACE Now, we discuss the strong and weak convergence let (X,,...,,..., ) bean generalized n-inner product space and,,..., be the induced n-norm. A sequence x k in X is said to converge strongly to a point x X whenever x k x, a 2,...,a n 0 for every a 1,...,a n X. In such a case, we write x k x. Meanwhile, x k is said to converge weakly to x whenever x k x, a 2,...,a n b 1,...,b n 0 for every a 2,...,a n,b 1,...,b n X. Clearly if x k and y k converges strongly/weakly to x and y respectively, then, for any α, β R, αx k + βy k converges strongly/weakly to αx + βy. Now, we show that a sequence cannot converges weakly to two distinct point as follows: Proposition 3.1. If x k converges weakly to x and x simultaneously, then x = x. Proof. By hypothesis and Property (3.5) of generalized n-inner products, we have x k,a 2,...,a n b 1,...,b n x, a 2,...,a n b 1,...,b n and x k,a 2,...,a n b 1,...,b n x,a 2,...,a n b 1,...,b n for every a 2,...,a n,b 1,...,b n X. By the uniqueness of the limit of a sequence of real numbers, we must have x, a 2,...,a n b 1,...,b n = x,a 2,...,a n b 1,...,b n x x,a 2,...,a n b 1,...,b n =0 for every a 2,...,a n,b 1,...,b n X. In particular, by taking b 1 = x x and a i = b i for each i =2,...,n. We obtain x x,a 2,...,a n = 0 for every a 2,...,a n X. By property (2.1) of n-norms and elementary linear algebra, this can only happen if x x =0orx = x. The next proposition says that the strong convergence implies the weak convergence in Generalized n-inner product spaces. Proposition 3.2. If x k converges strongly to x, then it converges weakly to x. Proof. By the Cauchy-Schwarz inequality we have x k x, a 2,...,a n b 1,b 2,...,b n x k x, a 2,...,a n x k x, a 2,...,a n b 1,b 2,...,b n b 1,b 2,...,b n x k x, a 2,...,a n b 1,b 2,...,b n for every a 2,...,a n,b 1,...,b n X. Since x k converges strongly to x,thus x k x,a 2,...,a n 0ask 0 for every a 2,...,a n X so, x k x, a 2,...,a n b 1,b 2,...,b n 0 for every a 2,...,a n X.

7 Results in generalized n-inner product spaces 1019 Corollary. A sequence cannot converge strongly to two distinct points. But weakly convergent sequences do not converge strongly, we give a counter example. Example 1. Let (X,.,. ) be a separable Hilbert space of infinite dimension and e k indexed by N, be an orthonormal basis for X. Then, for each x and b X, we have k x, e k x, b = x, b. In particular, if x = z, then we have parseval s identity k x, e k 2 = x 2 where =.,. 1 2 denotes the induced norm. Now X with the standard generalized n-inner product.,.,...,..,...,. Then, for each x, a 3,...,a n,b 1,...,b n X, we have the following identity x, b 1,...,b n 1 e k,b 1,...,b n 1 2 = x, e k b 1,...,b n 1 2 k k = x, b 1,...,b n 1 2 b 1,...,b n 1 2 n 1. Where.,.,...,. n 1 denotes the standard (n 1)-norm on X. For n = 2 we see, x, b e k,b 2 = x, e k b 2 k k = k [ x, e k b 2 x, b b, e k ] 2 = k [ x, e k 2 b 4 2 x, e k b, e k x, b b 2 + x, b 2 b, e k 2 ] = x 2 b 4 2 x, b 2 b 2 + x, b 2 b 2 = [ x 2 b 2 x, b 2 ] b 2 = x, b 2 b 2. Now, the counter example is e k. Beacause of Parseval s identity, we must have x, b 1,...,b n e k,b 1,...,b n 0 for every x, b 1,...,b n X. i.e, e k converges weakly to 0. Now for k N and b 1,...,b n X denote by e k the orthogonal projection of e k on spanned by b 1,...,b n. Then one may observe that e k e k 1. Hence e k,b 1,...,b n = e k e k b 1,...,b n n 1 b 1,...,b n n 1 0 where b 1,...,b n 1 are linearly independent. This shows e k does not converge strongly to 0inX. But in the finite dimensional case, both convergence are equivalent. 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 M. MASHADI, On n-normed spaces, Int. J. Math. Math. Sci. (2001),

8 1020 R. Chugh and Sushma [4] H. GUNAWAN, On n-inner products, n-norms, and the Cauchy-Schwarz inequality, Sci. Math. Jpn. 5(2002), [5] K. TRENCEVSKI and R. MALCESKI, On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality, J. Inequal. Pure and Appl. Math. 7(2006). Received: September, 2008