SOOCHOW JOURNAL OF MATHEMATICS Volume 28, No. 4, pp. 389-398, October 2002 INNER PRODUCTS ON n-INNER PRODUCT SPACES BY HENDRA GUNAWAN Abstract. In this note, we show that in any n-inner product space with n ≥ 2 we can explicitly derive an inner product or, more generally, an (n − k)-inner product from the n-inner product, for each k 2 f1; : : : ; n − 1g. We also present some related results on n-normed spaces. 1. Introduction Let n be a nonnegative integer and X be a real vector space of dimension d ≥ n (d may be infinite). A real-valued function h·; ·|·; : : : ; ·i on X n+1 satisfying the following five properties: (I1) hx1; x1jx2; : : : ; xni ≥ 0; hx1; x1jx2; : : : ; xni = 0 if and only if x1; x2; : : : ; xn are linearly dependent; (I2) hx1; x1jx2; : : : ; xni = hxi1 ; xi1 jxi2 ; : : : ; xin i for every permutation (i1; : : : ; in) of (1; : : : ; n); (I3) hx; yjx2; : : : ; xni = hy; xjx2; : : : ; xni; (I4) hαx, yjx2; : : : ; xni = αhx; yjx2; : : : ; xni; α 2 R; 0 0 (I5) hx + x ; yjx2; : : : ; xni = hx; yjx2; : : : ; xni + hx ; yjx2; : : : ; xni; is called an n-inner product on X, and the pair (X; h·; ·|·; : : : ; ·i) is called an n-inner product space. For n = 1, the expression hx; yjx2; : : : ; xni is to be understood as hx; yi, which denotes nothing but an inner product on X. The concept of 2-inner product spaces was first introduced by Diminnie, G¨ahler and White [2, 3, 7] in 1970's, Received March 26, 2001; revised August 13, 2002. AMS Subject Classification. 46C50, 46B20, 46B99, 46A19. Key words. n-inner product spaces, n-normed spaces, inner product spaces. 389 390 HENDRA GUNAWAN while its generalization for n ≥ 2 was developed by Misiak [12] in 1989. Note here that our definition of n-inner products is slightly simpler than, but equivalent to, that in [12]. On an n-inner product space (X; h·; ·|·; : : : ; ·i), one may observe that the following function 1=2 kx1; x2; : : : ; xnk := hx1; x1jx2; : : : ; xni ; defines an n-norm, which enjoys the following four properties: (N1) kx1; : : : ; xnk ≥ 0; kx1; : : : ; xnk = 0 if and only if x1; : : : ; xn are linearly dependent; (N2) kx1; : : : ; xnk is invariant under permutation; (N3) kαx1; x2; : : : ; xnk = jαj kx1; x2; : : : ; xnk; α 2 R; (N4) kx + y; x2; : : : ; xnk ≤ kx; x2; : : : ; xnk + ky; x2; : : : ; xnk: Just as in an inner product space, we have the Cauchy-Schwarz inequality: jhx; yjx2; : : : ; xnij ≤ kx; x2; : : : ; xnk ky; x2; : : : ; xnk; and the equality holds if and only if x; y; x2; : : : ; xn are linearly dependent (see [9]). Furthermore, we have the polarization identity: 2 2 kx + y; x2; : : : ; xnk − kx − y; x2; : : : ; xnk = 4hx; yjx2; : : : ; xni; and the parallelogram law: 2 2 2 2 kx+y; x2; : : : ; xnk +kx−y; x2; : : : ; xnk = 2(kx; x2; : : : ; xnk +ky; x2; : : : ; xnk ): The latter gives a characterization of n-inner product spaces. By the polarization identity and the property (I2), one may observe that hx; yjx2; : : : ; xni = hx; yjxi2 ; : : : ; xin i; for every permutation (i2; : : : ; in) of (2; : : : ; n). Moreover, one can also show that hx; yjx2; : : : ; xni = 0; when x or y is a linear combination of x2; : : : ; xn, or when x2; : : : ; xn are linearly dependent. INNER PRODUCTS ON n-INNER PRODUCT SPACES 391 Now, for example, if (X; h·; ·i) is an inner product space, then the following function hx; yi hx; x2i · · · hx; xni hx ; yi hx ; x i · · · hx ; x i 2 2 2 2 n hx; yjx2; : : : ; xni := . ; . .. hx ; yi hx ; x i · · · hx ; x i n n 2 n n defines an n-inner product, called the standard (or simple) n-inner product on X. Its induced n-norm kx1; x2; : : : ; xnk represents the volume of the n-dimensional parallelepiped spanned by x1; x2; : : : ; xn. Historically, the concept of n-norms were introduced earlier by G¨ahler in order to generalize the notion of length, area and volume in a real vector space (see [4, 5, 6]). The objects studied here are n-dimensional parallelepipeds. The concept of n-inner products is thus useful when one talks about the angle between two n-dimensional parallelepipeds having the same (n − 1)-dimensional base. In this note, we shall show that in any n-inner product space with n ≥ 2 we can derive an (n − k)-inner product from the n-inner product for each k 2 f1; : : : ; n−1g. In particular, in any n-inner product space, we can derive an inner product from the n-inner product, so that one can talk about, for instance, the angle between two vectors, as one might like to. In addition, we shall present some related results on n-normed spaces. See [5] and [11] for previous results on these spaces. 2. Main Results To avoid confusion, we shall sometimes denote an n-inner product by h·; ·|·; : : : ; ·in and an n-norm by k·; : : : ; ·kn. Let (X; h·; ·|·; : : : ; ·in) be an n-inner product space with n ≥ 2. Fix a linearly independent set fa1; : : : ; ang in X. With respect to fa1; : : : ; ang, define for each n−k+1 k 2 f1; : : : ; n − 1g the function h·; ·|·; : : : ; ·in−k on X by hx; yjx2; : : : ; xn−ki := hx; yjx2; : : : ; xn−k; ai1 ; : : : ; aik i: fi1;:::;ikX}⊆{1;:::;ng Then we have the following fact: 392 HENDRA GUNAWAN Fact 2.1. For every k 2 f1; : : : ; n − 1g, the function h·; ·|·; : : : ; ·in−k defines an (n − k)-inner product on X. In particular, when k = n − 1, hx; yi := hx; yjai2 ; : : : ; ain i; (1) fi2;:::;inX}⊆{1;:::;ng defines an inner product on X. Proof. It is not hard to see that the function h·; ·|·; : : : ; ·in−k satisfies the five properties (I1){(I5) of an (n − k)-inner product, except perhaps to estab- lish the second part of (I1). To verify this property, suppose that x1; : : : ; xn−k are linearly dependent. Then hx1; x1jx2; : : : ; xn−k; ai1 ; : : : ; aik i = 0 for every fi1; : : : ; ikg ⊆ f1; : : : ; ng, and hence hx1; x1jx2; : : : ; xn−ki = 0. Conversely, sup- pose that hx1; x1jx2; : : : ; xn−ki = 0. Then hx1; x1jx2; : : : ; xn−k; ai1 ; : : : ; aik i = 0 for every fi1; : : : ; ikg ⊆ f1; : : : ; ng. Hence fx1; : : : ; xn−k; ai1 ; : : : ; aik g are linearly dependent for every fi1; : : : ; ikg ⊆ f1; : : : ; ng. By elementary linear algebra, this can only happen if x1; : : : ; xn−k are linearly dependent (or if x1 = 0 when k = n − 1). Corollary 2.2. Any n-inner product space is an (n − k)-inner product space for every k = 1; : : : ; n − 1. In particular, an n-inner product space is an inner product space. Corollary 2.3. Let k·; : : : ; ·kn be the induced n-norm on X. Then, for each k 2 f1; : : : ; n − 1g, the following function 1=2 2 kx1; : : : ; xn−kk := kx1; : : : ; xn−k; ai1 ; : : : ; aik k ; fi1;:::;ikX}⊆{1;:::;ng defines an (n − k)-norm that corresponds to h·; ·|·; : : : ; ·in−k on X. In particular, 1=2 2 kxk := kx; ai2 ; : : : ; ain k ; fi2;:::;inX}⊆{1;:::;ng defines a norm that corresponds to the derived inner product h·; ·i on X. Note that by using a derived inner product, one can develop the notion of orthogonality and the Fourier series theory in an n-inner product space, just INNER PRODUCTS ON n-INNER PRODUCT SPACES 393 like in an inner product space (see [3] and [13] for previous results in this direc- tion). With respect to the derived inner product h·; ·i defined by (1), one may observe that the set fa1; : : : ; ang is orthogonal and that kaik = ka1; : : : ; ank for every i = 1; : : : ; n (see [8]). In particular, if X is n-dimensional, then fa1; : : : ; ang forms an orthogonal basis for X and each x 2 X can be written −2 n as x = ka1; : : : ; ank i=1hx; aiiai. Unlike in [13], wePcan now have an orthogonal set of m vectors with 1 ≤ m < n. In general, by using a derived inner product, we have a more relaxed condition for orthogonality than that in [3] or [13]. Furthermore, one may also use the derived inner products and their induced norm to study the convergence of sequences of vectors in an n-inner product space. See some recent results in [10]. 2.1. Related results on n-normed spaces Suppose now that (X; k·; : : : ; ·kn) is an n-normed space and, as before, fa1; : : : ; ang is a linearly independent set in X. Then one may check that for each k 2 f1; : : : ; n − 1g 1=2 2 kx1; : : : ; xn−kk := kx1; : : : ; xn−k; ai1 ; : : : ; aik k ; (2) fi1;:::;ikX}⊆{1;:::;ng defines an (n − k)-norm on X (see [5] and [11] for similar results). In particular, the triangle inequality can be verified as follows: kx + y; x2; : : : ; xn−kk 1=2 2 = kx + y; x2; : : : ; xn−k; ai1 ; : : : ; aik k fi1;:::;ikX}⊆{1;:::;ng ≤ kx; x2; : : : ; xn−k; ai1 ; : : : ; aik k fi1;:::;ikX}⊆{1;:::;ng 2 1=2 +ky; x2; : : : ; xn−k; ai1 ; : : : ; aik k 1=2 2 ≤ kx; x2; : : : ; xn−k; ai1 ; : : : ; aik k fi1;:::;ikX}⊆{1;:::;ng 394 HENDRA GUNAWAN 1=2 2 + ky; x2; : : : ; xn−k; ai1 ; : : : ; aik k fi1;:::;ikX}⊆{1;:::;ng = kx; x2; : : : ; xn−kk + ky; x2; : : : ; xn−kk: The first inequality follows from the triangle inequality for the n-norm, while the second one follows from the triangle inequality for the l2-type norm. In general, for 1 ≤ p ≤ 1, one may observe that 1=p p kx1; : : : ; xn−kk := kx1; : : : ; xn−k; ai1 ; : : : ; aik k ; fi1;:::;ikX}⊆{1;:::;ng also defines an (n − k)-norm on X. Among these derived (n − k)-norms, however, the case p = 2 is special in the following sense.