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On a Generalized $n-$inner Product and the Corresponding Cauchy-Schwarz Inequality  
 
  Authors: Kostadin Trencevski, Risto Malcevski,  
  Keywords: Cauchy-Schwarz inequality, $n$-inner product, $n$-norm.  
  Date Received: 22/11/04  
  Date Accepted: 27/02/06  
  Subject Codes:

46C05, 26D20.

  
  Editors: Constantin P. Niculescu,  
 
  Abstract:

In this paper is defined an $n$-inner product of type $langle mathbf{a}_1,dots ,mathbf{a}_nvert mathbf{b}_1cdots mathbf{b}_nrangle $ where $mathbf{a}_1,dots ,mathbf{a}_n$, $mathbf{b}_1, dots ,mathbf{b}_n$ are vectors from a vector space $V$. This definition generalizes the definition of Misiak of $n$-inner product [2], such that in special case if we consider only such pairs of sets ${ mathbf{a}_1,dots ,mathbf{a}_1}$ and ${ mathbf{b}_1cdots mathbf{b}_n}$ which differ for at most one vector, we obtain the definition of Misiak. The Cauchy-Schwarz inequality for this general type of $n$-inner product is proved and some applications are given.;


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