## 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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