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n-inner product spaces. (English) Zbl 0673.46012

Let n be a natural number \(\geq 1\), L a linear space of dimension \(\geq n\) and \((\cdot,\cdot | \cdot,...,\cdot)\) a real-valued function on \(L^{n+1}\) satisfying the following six conditions:
1. \((a,a| a_ 2,...,a_ n)\geq 0\), and the equality to 0 takes place if and only if \(a,a_ 2,...,a_ n\) are linearly dependent.
2. \((a,b| a_ 2,...,a_ n)=(b,a| a_ 2,...,a_ n).\)
3. \((a,b| a_ 2,...,a_ n)=(a,b| a_{i_ 2},...,a_{i_ n})\) for every permutation \((i_ 2,...,i_ n)\) of (2,...,n).
4. If \(n>1\) \((a,a| a_ 2,...,a_ n)=(a_ 2,a_ 2| a,a_ 3,...,a_ n).\)
5. \((\alpha a,b| a_ 2,...,a_ n)=\alpha (a,b| a_ 2,...,a_ n)\) for \(\alpha\in {\mathbb{R}}.\)
6. \((a+a',b| a_ 2,...,a_ n)=(a,b| a_ 2,...,a_ n)+(a',b| a_ 2,...,a_ n).\)
The function (\(\cdot,\cdot | \cdot,...,\cdot)\) is called an n-inner product on L and L equipped with the n-inner product is said to be an n- inner product space. This generalizes the notion of inner product space \((n=1)\) and the author shows that the Cauchy-Buniakowski inequality, namely \(| (a,b| a_ 2,...,a_ n)| \leq \sqrt{(a,a| a_ 2,...,a_ n)(b,b| a_ 2,...,a_ n)}\) is valid also for n-inner product spaces. Analogously he introduces n-norms on L as a function \(\| \cdot,...,\cdot \|\) on \(L^ n\) with values in \({\mathbb{R}}^+\) and shows that for any n-inner product (\(\cdot,\cdot | \cdot,...,\cdot)\) on L the function \(\| \cdot,...,\cdot \|\) defined by \(\| a_ 1,a_ 2,...,a_ n\| =\sqrt{(a_ 1,a_ 1| a_ 2,...,a_ n)}\) is an n-norm on L.
Let \({\mathcal N}'_ L\) denote the collection of all formal sums \(\sum^{m}_{i=1}a^ i_ 1\times a^ i_ 2\times...\times a^ i_ n\) where \(a^ i_ k\) \((k=1,...,n\); \(i=1,...,m)\) are elements of L and it is assumed as before that dim \(L\geq n\). Two elements in \({\mathcal N}'_ L\), \(\sum^{m}_{i=1}a^ i_ 1\times...\times a^ i_ n\) and \(\sum^{m'}_{i=1}b^ i_ 1\times...\times b^ i_ n\) are said to be equivalent if for any linear functionals \(f_ j\) \((j=1,...,n)\) we have \[ \sum^{m}_{i=1}\left| \begin{matrix} f_ 1(a^ i_ 1),f_ 2(a^ i_ 1),...,f_ n(a^ i_ 1)\\ f_ 1(a^ i_ n),f_ 2(a^ i_ n),...,f_ n(a^ i_ n) \end{matrix} \right| = \sum^{m'}_{i=1} \left| \begin{matrix} f_ 1(b^ i_ 1),...,f_ n(b^ i_ 1) \\ f_ 1(b^ i_ n),...,f_ n(b^ i_ n)\end{matrix} \right|. \] The quotient space arising from \({\mathcal N}'_ L\) modulo this equivalence relation is denoted by \({\mathcal N}_ L\) and its elements are called n-vectors. The author shows that if \((\cdot,\cdot)\) is an inner product on \({\mathcal N}_ L\) then the function \((\cdot,\cdot | \cdot,...,\cdot)\) defined on \(L^{n+1}\) by \((a,b| a_ 2,...,a_ n)=({\mathfrak n}(a\times a_ 2\times...\times a_ n),{\mathfrak n}(b,a_ 2,...,a_ n))\) (where \({\mathfrak n}(a_ 1\times...\times a_ n)\) denotes the class in \({\mathcal N}_ L\) containing the element \(a_ 1\times...\times a_ n\) in \({\mathcal N}'_ L)\) defines an n-inner product on L.
Finally, he introduces the concept of an n-metric \(\sigma\) on an abstract set X as a generalization of a metric \((n=1)\) and shows that X carries a certain topology naturally induced by \(\sigma\) on X. He also shows that if the linear space L is equipped with the n-norm \(\| \cdot,...,\cdot \|\) then L is also an n-metric space via the formula \(\sigma (a_ 1,...,a_{n+1})=\| a_ 1-a_{n+1},...,a_ n-a_{n+1}\|\).
Reviewer: L.Janos

MSC:

46C99 Inner product spaces and their generalizations, Hilbert spaces
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References:

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