The Generalization of Apollonious Identity to Linear n-normed Spaces

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1 Int. J. Contemp. Math. Sciences, Vol. 5, 010, no. 4, The Generalization of Apollonious Identity to Linear n-normed Spaces Mehmet Açıkgöz University of Gaziantep Faculty of Science and Arts Department of Mathematics 7310 Gaziantep, Turkey Nurgül Aslan University of Gaziantep Faculty of Science and Arts Department of Mathematics 7310 Gaziantep, Turkey Serkan Aracı University of Gaziantep Faculty of Science and Arts Department of Mathematics 7310 Gaziantep, Turkey Abstract. We shall extend the well known Apollonious identity in functional analysis to linear n-normed spaces and prove it in this spaces. Mathematics Subject Classification: 41A65, 46A15, 46B05, 46B0 Keywords: n-inner product, n-normed space, Apollonious identity 1. Introduction In [4] and [5], S. Gahler introduced an attractive theory of -norm and n- norm on a linear space. Since then these were studied in many papers such as [3] and [7].

2 1188 M. Açıkgöz, N. Aslan and S. Aracı In [6], H. Gunawan and M. Mashadi gave a simple way to derive an (n 1)- norm from the n-norm and realized that any n-normed space is an (n 1)- normed space. Generalized n-inner product spaces defined by K. Trencevski and R. Malceski [9] which are generalization of n-inner product spaces introduced by A. Misiak [8]. Definition 1. Let n N 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 n =0iffx 1,,x n are linearly dependent, 1.) x 1,,x n is invariant under any permutation, 1.3) x 1,,x n 1,αx n = α x 1,,x n 1,x n for any α R, 1.4) x 1,,x n 1,y+ z x 1,,x n 1,y + x 1,,x n 1,z is called an n-norm on X and the pair (X,.,...,. ) is called n-normed linear space. A trivial example of an n-normed space is X = R n equipped with the following n-norm; x 1,,x n E = abs x x 1n x n1... x nn where x i =(x i1,,x in ) R n for each i =1,,,n (The subscript E is for Euclidean). The properties of n-normed spaces can be given as follows: P1) x 1,,x n 1,x n 0, P) x 1,,x n 1,x n = x 1,,x n 1,x n + α 1 x α n 1 x n 1 for every x 1,x,,x n X and α 1,,α n 1 R. Definition. [] Assume that n is a positive integer and X is a real linear space such that dim X n and.,..,...,. is a real function defined on X X X = X n+1 such that.1) x 1,x 1 x,..., x n 0, for any x 1,x,,x n X and x 1,x 1 x,..., x n = 0 if and only if x 1,x,,x n are linearly dependent vectors,.) a, b x 1,..., x n 1 = ϕ (a),ϕ(b) π (x 1 ),..., π (x n 1 ), for any a, b, x 1,x,,x n 1 X and for any bijection π : {x 1,x,,x n 1 } {x 1,x,,x n 1 } and ϕ : {a, b} {a, b},.3) If n>1, then x 1,x 1 x,..., x n = x,x x 1,x 3,..., x n for any x 1,x,,x n X.

3 Apollonious identity to linear n-normed spaces ) αy, z x 1,..., x n 1 = α y, z x 1,..., x n 1 for any y, z, x 1,x,,x n 1 X and any scalar α R..5) y + y 1,z x 1,..., x n 1 = y, z x 1,..., x n 1 + y 1,z x 1,..., x n 1 for any y, y 1,z,x 1,,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 n = x 1,x 1 x,..., x n. Definition 3. [] Assume that n is a positive integer, X is a real linear space such that dim X n and.,...,..,...,. is a real function on X n such that 3.1) x 1,..., x n x 1,..., x n 0ifx 1,..., x n are linearly independent vectors, 3.) x 1,..., x n y 1,..., y n = y 1,..., y n x 1,..., x n for x 1,..., x n,y 1,..., y n X, 3.3) λx 1,..., x n y 1,..., y n = λ x 1,..., x n y 1,..., y n for any scalar λ R and x 1,..., x n,y 1,..., y n X, 3.4) x 1,..., x n y 1,..., y n = x σ(1),..., x σ(n) y 1,..., y n for any odd permutation σ in the set {1,,..., n} and x 1,..., x n,y 1,..., y n X, 3.5) x 1 + z, x,..., x n y 1,..., y n = x 1,x,..., x n y 1,..., y n + z, x,..., x n y 1,..., y n for any x 1,..., x n,y 1,..., y n,z X, 3.6) If x 1,y 1,..., y i 1,y i+1,..., y n y 1,..., y n = 0 for each i {1,,..., n}, then x 1,..., x n y 1,..., y n = 0 for arbitrary vectors x,..., x n. Then the function.,...,..,...,. is called a generalized n-inner product space. In the special case, if we consider only such pairs of sets x 1,..., x n and y 1,..., y n which differ for at most one vector, for example x 1 = x, y 1 = y and x = y = a 1,,x n = y n = a n 1 then by putting x, y a 1,..., a n 1 = x, a 1,..., a n 1 y, a 1,..., a n 1 we obtain an n-inner product. Example 1. [9] Let X be a space with inner product.. then a 1 b 1 a 1 b... a 1 b n a b 1 a b... a b n... a 1,..., a n b 1,..., b n = a n b 1 a n b... a n b n defines a generalized n-inner product on X. Proposition 1. Let.,...,. be a n-norm on X. Then x + y, x,,x n + x y, x,,x n (1.1) = x, x,,x n + y, x,,x n where x, y, x,x 3,,x n X.

4 1190 M. Açıkgöz, N. Aslan and S. Aracı Proof. (See [].) The equality given in (1.1) is called parallelogram law in generalized n-inner product space.. Main Results In this section, for our main result we shall define the Apollonious identity in linear n-normed spaces, and give its proof in this spaces. For n =,we obtain our results in [1]. Definition 4. Let X be an n-inner product. For x, y, z X and x,,x n X, the Apollonius identity x y, x,,x n + x z, x,,x n = 1 y z, x,,x n + x y + z,x (.1),,x n where.,...,. =.,..,...,.. Theorem 1. The identity (.1) is true. Proof. To prove this identity; let z = y, then we have x y, x,,x n + x + y, x,,x n = 1 y, x,,x n + x, x,,x n which is known as parallelogram law. Now, let us use the definition and properties of n-inner product to prove the Apollonious identity, we have x y, x,,x n + x z, x,,x n = x y, (x y) x,,x n + x z, (x z) x,,x n = x, (x y) x,,x n y, (x y) x,,x n + x, (x z) x,,x n z, (x z) x,,x n = x, x x,,x n + y, y x,,x n + z, z x,,x n x, y x,,x n x, z x,,x n y, x x,,x n z, x x,,x n = 1 [ y, (y z) x,..., x n z, (y z) x,..., x n [ ( + x, x y + z ) y + z x,..., x n, = 1 [ [ y z, (y z) x,..., x n + x y + z, = 1 y z, x,..., x n + x y + z,x,..., x n which is the desired result. ( x y + z ( x y + z ) x,..., x n ) x,..., x n

5 Apollonious identity to linear n-normed spaces 1191 Theorem. The identity x y, x,..., x n + x z, x,..., x n is true. = 1 y z, x,..., x n + x, x,..., x n Proof. By using the above fact, we can write x y, x,..., x n + x z, x,..., x n = x y, x,..., x n x y, x,,x n + x z, x,..., x n x z, x,,x n = x, x,..., x n x y, x,,x n y, x,..., x n x y, x,,x n + x, x,..., x n x z, x,,x n z, x,..., x n x z, x,,x n = x, x,..., x n x, x,,x n x, x,..., x n y, x,,x n y, x,..., x n x, x,,x n + y, x,..., x n y, x,,x n + x, x,..., x n x, x,,x n x, x,..., x n z, x,,x n z, x,..., x n x, x,,x n + z, x,..., x n z, x,,x n = x, x,..., x n x, x,,x n + y, x,..., x n y, x,,x n + z, x,..., x n z, x,,x n x, x,..., x n y, x,,x n x, x,..., x n z, x,,x n y, x,..., x n x, x,,x n z, x,..., x n x, x,,x n = 1 [ y, x,..., x n y z, x,,x n z, x,..., x n y z, x,,x n ( +[ x, x,..., x n x y + z ),x,,x n ( y + z,x,..., x n x y + z ),x,,x n = 1 [ y z, x,..., x n (y z),x,,x n +[ x y + z (,x,..., x n x y + z ),x,,x n = 1 y z, x,..., x n + x y + z,x,..., x n which is the desired result. References [1] Acikgoz, M., Karakuş, Y., et all, Apollonious identity in linear -normed spaces, Numerical Analysis and Applied Mathematics, Int. Conference on Applied Mathematics 009: Vol. 1 and Vol. ; doi: / [] Chugh, R., and Sushma, Some results in generalized n-inner product spaces, Inter. Math. Forum. 4, 009, no. 1,

6 119 M. Açıkgöz, N. Aslan and S. Aracı [3] Diminie, C., and White, A.G., Non expansive mappings in linear -normed spaces, Math. Japonica 1(1976), [4] Gahler, S., Lineare -normierte Raume, Math.Nachr., 8 (1964), [5] Gahler, S., Untersuchungen uber verallgemenerte, m-metrische Raume, I, II, III, Math. Nachr. 40(1969), [6] Gunawan, H., Mashadi, M., On n-normed space, Int. J. Math. Sci. 7(001), No. 10, [7] Iseki, K., On non-expansive mapping in strictly convex linear -normed spaces, Math. Sem. Note Kobe University 3(1975), [8] Misiak, A., n- inner product spaces, Math. Nachr. 140(1989), [9] Trencevski, K., Malceski, R., On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality, Journal Inequality Pure and Appl. Math. 7(007). Received: November, 009