IntrodrictioitIn 1928 K. MENGER made considerations concerning axioms for an abstract n-dmensional metric ([lo]). The essential development in the theory of higherdimensional metrics, however, began in 1962 where S. GXHLER ([4], [5]) introduced practicable definitions of a %metric and a 2-normed space end started with systematic investigations of these spaces, above all concerning their topological properties. Later these spaces have been used by several authors for the'study of geometrical, analytical and functional analytical questions. Of especial importance are the considerations concerning %inner product spaces ([l], [ 2 ] , [3], [9], [ll]).The results of ([4], [5]) on ?-metric and 2-normed spaces were extended by S. Giirr-LER to n-metric and n-normed spaces, where n is an arbitrary iistural number L 1. The very interesting results made it desirable to stud? z-inner product spaces with arbitrary natural n z 1. This paper concerns with fundamental properties of n-inner products, among it gives a formula, which is an extension of the CAUCHY-BUNISEOWSEI Inequality to arbitrary n. There are studied relations between n-inner products and certain partially symmetric covariant tensors of rank 27% as Tell as relations between n-inner products and n-norms, too.2. , ..., .) we also write (-, .) and (a, b I a2, ..., an) is to be anderstood as the ex-+) This article is a part of the dissertation (Technical University Warsaw) which was written under supervision of Siegfried Giihler.

Abstract. In this paper we show that at most 2 gcd(m, n) points can be placed with no three in a line on an m × n discrete torus. In the situation when gcd(m, n) is a prime, we completely solve the problem.

Introduction This paper is a continuation of investigations an n-inner product spaces given in [S]and an extension of results given in [2] and [5] to arbitrary natural n.Let n be a natural number $: 0, L be a linear space of dimension 2 n and (-, -I ., . . ., .)be a real function on L"+1. In the case n = 1 instead of ( a , . I ., ..., .) we also write (-, a) and (a, b I a,, . . ., a,) is to be understood as the expression (a, b). Let us assume the following conditions: 1. (a,ala,,...,a,) 20, 2. (a, b I 4, (a, a I az, ..., a,) = 0 if and only if a, a*, . .., a, are linearly dependent. * -2 a,) = (b, a I a,, ...) am)-3. (a, b I az, ..., a,) = (a, b I ai,, . . ., ai,) for every permutation (i,, . . ., in) of (2, .. ., n). 4. If n > 1, (a, a I a,, 4, . .., a,) = (a,, a, I a, a3, . . ., %I. 5. (aa, b I a,, ..., a,) = u(a, b I a,, ..., a,) for every real u. 6. (a + a', b I a,, . . ., a,) = (a, b I up, . . ., a,) + (a', b I a,, .. ., am). Then (-, . I ., . . ., .) is called an n-inner product on L and L equipped with the n-inner product is said to be an n-inner product s p e ([S]). The concept of an n-inner product space is a generalization of the concepts of an inner product space (n = I) and of a 2-inner product space ([l], [5]). Every n-inner product apace will be considered to be an n-normed space ([3], [S]) equipped with the n-norm IIa1, a,, * --2 an11 = f(.i, a1 I 4, * --9 am) -Every n-normed space has a natural topology defined by ita n-norm ([4], [S]). This topology on L is locally convex and separated. An n-inner product space d W 8 y S be considered to be a topological space equipped with thie topology. In the m e that L possessea property K the natural topology of L agrees with the topology given by the norm where bl, ..., b. are arbitrary elements of L with Ilbl, ..., b, II =I= 0. *) This article is 8 pert of the dissertation (Technical University Warsaw).

REMARKS ON 2-INNER PRODUCTS

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