ON GENERALIZED n-inner PRODUCT SPACES

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1 Novi Sad J Math Vol 41, No 2, 2011, ON GENERALIZED n-inner PRODUCT SPACES Renu Chugh 1, Sushma Lather 2 Abstract The primary purpose of this paper is to derive a generalized (n k) inner product with n 2, from the generalized n-inner product, which is a generalization of the definition of Misiak [3] of the n-inner product for each k {1, 2, n 1} and also provide results related to the n-normed product induced by generalized n-inner product AMS Mathematics Subject Classification (2010): Primary 46C05, 46C99; Secondary 26D15, 26D10 Key words and phrases: n-norm linear space, n-inner product space, Cauchy-Schwarz inequality, Polarization Identity, Parallelogram law, generalized n-inner product space 1 Introduction Misiak [3] has introduced an n-norm and n-inner product by the following definitions Definition 11 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 n satisfying the following four properties: (i) x 1, x 2,, x n 0 if any only if x 1, x 2,, x n are linearly dependent, (ii) x 1, x 2,, x n is invariant under any permutation, (iii) x 1, x 2,, ax n a x 1, x 2,, x n, for any a R (real), (iv) 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 Definition 12 Assume that n is a positive integer and X is a real vector space such that dim X n and (,,, ) is a real function defined on X n+1 such n 1 that: (i) (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; (ii) (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 2,, x n 1 X and for any bijections π : {x 1, x 2,, x n 1 } {x 1, x 2,, x n 1 } and φ : {a, b} {a, b}; 1 Department of Mathematics, MD University, Rohtak, India 2 Department of Mathematics, MD University, Rohtak, India, lathersushma@yahoocom

2 74 Renu Chugh, Sushma Lather (iii) 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; (iv) (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; (v) (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 n 1 X Then (,,, ) is called n-inner product and (X, (,,, ) n-prehilbert n 1 n 1 space If n 1, then Definition 12 reduces to the ordinary inner product This n-inner product induces an n-norm [3] by x 1,, x n (x 1, x 1 x 2,, x n ) Trencevski and Malceski [4] gave the definition of generalized n-inner product and the Cauchy-Schwarz inequality in this space as Definition 13 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 (I 1 ) a 1,, a n a 1,, a n > 0 if a 1,, a n are linearly independent vectors, (I 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 (I 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, (I 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, (I 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, (I 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 1,, a n Then the function,,,, is called generalized n-inner product and the pair (X,,,,, ) is called generalized n-prehilbert space The generalized n-inner product on X induces an n-norm [3] by x 1,, x n x 1,, x n x 1,, x n And Cauchy-Schwarz inequality in generalized n-inner product on X is given 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 In [1] we obtain the following identities:

3 On generalized n-inner product spaces 75 Polarization identity in generalized n-inner product space as 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 And parallelogram law in generalized n-inner product space as 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 2 The classical known example [4] of generalized n-inner product space is Example 14 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 a generalized n-inner product on X Misiak [3] generalized the definition of 2-inner product given by Gahler [4] in n-inner product Recently, Trencevski and Malceski [4] introduced the concept of generalized n-inner product as the generalization of n-inner product and obtained some related results In [1], we discussed the weak and strong convergence, and proved some identities in this space In this paper, we present a simple method to derive a generalized (n k) inner product with n 2, from the generalized n-inner product for each k {1, 2, n 1} and also provide results related to n-norm induced by generalized n-inner product The notion of orthogonality in a generalized n-inner product space can be developed by using a derived generalized inner product or inner product, just as in [1, 2, 4] 2 Main results To avoid confusion, we shall sometimes denote a generalized n-inner product by,,,,,, n and an n-norm by,,, n Theorem 21 Let (X,,,,,,, n ) be generalized n-inner product space with finite dimension d n 2 Take a linearly independent set {a 1, a 2,, a d } and define the following function,,,,, n 1 on X 2(n 1) by (21) x 1, x 2,, x n 1 y 1, y 2,, y n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i such that this function satisfies (I 6 ), then the function,,,,,, n 1 is a generalized (n 1)-inner product on X

4 76 Renu Chugh, Sushma Lather Proof We will verify that,,,,,, n 1 satisfies the following six properties of a generalized (n 1)-inner product (i) To verify this property, suppose that x 1, x 2,, x n 1 are linearly dependent Then x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0, for every i {1, 2,, d} and hence x 1, x 2,, x n 1 x 1, x 2,, x n 1 0 Conversely, suppose that then x 1, x 2,, x n 1 x 1, x 2,, x n 1 0, n x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0 so x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0 for each i {1, 2,, d} Hence by (I 1 ) x 1, x 2,, x n 1, a i are linearly dependent for each i {1, 2,, d} By elementary linear algebra, this can only happen if x 1, x 2,, x n 1 are linearly dependent (ii) By using (I 2 ), we have x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1, a i y 1, y 2,, y n 1 x 1, x 2,, x n 1 n 1 (iii) λx 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 λx 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i λ x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i λ x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 For any scalar λ R, using (I 3 )

5 On generalized n-inner product spaces 77 (iv) x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x σ(1), x σ(2),, x σ(n 1), a i y 1, y 2,, y n 1, a i x σ(1), x σ(2),, x σ(n 1) y 1, y 2,, y n 1 for any odd permutation σ in the set {1,, n} and using (I 4 ) (v) x 1 + z, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1 + z, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i + z, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 + z, x 2,, x n 1 y 1, y 2,, y n 1 n 1 (vi) If x 1, y 1,, y j 1, y j+1,, y n 1 y 1, y 2,, y n 1 n 1 {1, 2,, n 1} 0 for each j x 1, y 1,, y j 1, y j+1,, y n 1, a i y 1, y 2,, y n 1, a i 0, hence by the orthonormal basis {a 1, a 2,, a d } and assumption of the theorem, we have the required condition that x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 0 for arbitrary vectors x 2,, x n 1 So,,,,,,, n 1 is a generalized (n 1)-inner product on X Corollary 22 Every generalized n-inner product space is generalized (n k)- inner product space for all k 1, 2,, n 1, by induction with generalized (n k)-inner product x 1, x 2,, x n k y 1, y 2,, y n x 1, x 2,, x n k, a i1, a i2,, a ik y 1, y 2,, y n k, a i1, a i2,, a ik n i 1,i 2,,i k {1,2,,d} such that this function satisfies (I 6 ), this condition is necessary for k 1, 2,, n 2, but for k n 1 it is trivially satisfied In particular,

6 78 Renu Chugh, Sushma Lather every generalized n-inner product space induces an inner product space ie x, y x, a i1, a i2,, a in 1 y, a i1, a i2,, a in 1 n i 1,i 2,,i n 1 {1,2,,d} Corollary 23 Let,,, n be the induced n-norm from a generalized n-inner product on X Then the following function ( ) 1 x 1, x 2,, x n 1 n 1 x 1, x 2,, x n 1, a i 2 2 is an (n 1)-norm that corresponds to,,,,,, n 1 on X, and by induction we have x 1, x 2,, x n k n k ( In particular, x i 1,i 2,,i k 1,2,,d ( x 1, x 2,, x n k, a i1, a i2,, a ik 2 ) 1 2 i 1,i 2,,i k 1 1,2,,d x, a i1, a i2,, a in 1 2 ) 1 2 defines a norm that corresponds to the derived generalized inner product or inner product, on X 24 Related results on n-normed space induced from generalized n-inner product space Suppose now that (X,,,, n ) is an n-normed space and {a 1, a 2,, a d } is a linearly independent orthonormal set in X Then we can show that ( ) 1 x 1, x 2,, x n 1 n 1 x 1, x 2,, x n 1, a i 2 2 defines an (n 1)-norm on X verified as: In particular, the triangle inequality can be x + y, x 2,, x n 1 2 n 1 x + y, x 2,, x n 1 x + y, x 2,, x n 1 x + y, x 2,, x n 1, a i x + y, x 2,, x n 1, a i x + y, x 2,, x n 1, a i 2 ( x, x 2,, x n 1, a i + y, x 2,, x n 1, a i ) 2

7 On generalized n-inner product spaces 79 Thus x + y, x 2,, x n 1 n 1 ( ) 1 ( x, x 2,, x n 1, a i + y, x 2,, x n 1, a i ) 2 2 ( ) 1 ( x, x 2,, x n 1, a i 2 2 ) 1 + y, x 2,, x n 1, a i 2 2 x, x 2,, x n 1 n 1 + y, x 2,, x n 1 n 1 This inequality shows the triangle inequality in (n 1)-norm Theorem 25 If the n-norm induced by generalized n-inner product satisfies the parallelogram law 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 2, then the (n 1)-norm induced by generalized n-inner product given by (4) satisfies x + y, x 2,, x n x y, x 2,, x n 1 2 In particular, the derived norm satisfies 2 x, x 2,, x n y, x 2,, x n 1 2 x + y 2 + x y 2 2 x y 2 Proof Polarization Identity in a generalized n-inner product space is x, x 2,, x n y, x 2,, x n 1 4 ( x + y, x 2,, x n 2 x y, x 2,, x n 2 ) and a generalized (n 1)-inner product is derived from it with respect to {a 1, a 2,, a d } One will then realize that the derived (n 1)-norm is the induced (n 1)-norm from the derived generalized (n 1)-inner product, and hence the parallelogram law follows 3 Examples Example 31 Let X R n be equipped with the standard generalized n-inner product space x 1, y 1 x 1, y 2 x 1, y n x 2, y 1 x 2, y 2 x 2, y n x 1, x 2,, x n y 1, y 2,, y n x n, y 1 x n, y 2 x n, y n

8 80 Renu Chugh, Sushma Lather where x, y is the usual inner product on R n Then the derived generalized (n k) inner product with respect to an orthonormal basis {b 1, b 2,, b n } coincides with the standard generalized (n k)-inner product on R n, that is, x 1, x 2,, x n k y 1, y 2,, y n k x 1, y 1 x 1, y 2 x 1, y n k x 2, y 1 x 2, y 2 x 2, y n k x n k, y 1 x n k, y 2 x n k, y n k In particular, the derived generalized inner product (inner product) x, y with respect to {b 1, b 2,, b n }, which is given by is the usual inner product x, y x, b 2, b 3,, b n y, b 2, b 3,, b n + x, b 1, b 3,, b n y, b 1, b 3,, b n + + x, b 1, b 2,, b n 1 y, b 1, b 2,, b n 1 Example 32 Let X R d be equipped with the standard n-inner product as in (5), with x, y being the usual inner product on R d Then one may particularly observe that the derived inner product with respect to an orthonormal basis {b 1, b 2,, b d } is given by x, y x, b i2, b i3,, b in y, b i2, b i3,, b in {i 1,i 2,,i n } {1,2,,d} ( ) d 1 x, y, n 1 ( ) d 1 (d 1)! where n 1 (d n)!(n 1)! This derived inner product is better than the previous one in the sense that it is only a multiple of the usual inner product This example may also be extended to any finite d-dimensional inner product space X References [1] Chugh, R, Sushma, Some results in generalized n-inner product spaces International Mathematical Forum, 4 (2009), [2] Risteski, Ice B, Trencevski, KG, Principal values and principal subspaces of two subspaces of vector spaces with inner product Beitrage zur Alegbra and Geometric Contribution to Algebra and Geometry, Vol 42 No 1 (2001), [3] Misiak, A, n-inner product spaces Math Nachr 140 (1989), [4] Trencevski, K, Malceski, R, On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality J Inequal Pure and Appl Math, 7(2) Art 53, 2006 Received by the editors April 28, 2009

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