On the probabilistic inner product spaces
A new definition for probabilistic inner product spaces is given as follows: Definition. A probabilistic inner product space is a triplet (E,\({\mathcal F},\Delta)\), where E is a real linear space, \(\Delta\) is a t-norm and \({\mathcal F}\) is a mapping of \(E\times E\to {\mathcal D}\) \((F_{x,y}\) will denote the distribution function \({\mathcal F}(x,y)\), and \({\mathcal D}\) denotes the set of all left-continuous distribution functions) satisfying the following conditions: (1) \(F_{x,x}(0)=0\); (2) \(F_{x,y}=F_{y,x}\); (3) \(F_{x,x}(t)=H(t)\), \(\forall t\in R\) if and only if \(x=0\); where \(H(t)=1\), if \(t>0\), and \(=0\) if \(t\leq 1\); \[ (4)\quad F_{\alpha x,y}(t)=F_{x,y}(t/\alpha),\;if\;\alpha >0,\quad =H(t),\;if\;\alpha =0,\quad =1-F_{x,y}((t/\alpha)+)\;if\;\alpha <0, \] where \(\alpha\) is any real number, \(F_{x,y}((t/\alpha)+)\) is the right-hand limit of \(F_{x,y}\) at t/\(\alpha\) ; and \[ (5)\quad F_{x+y,z}(t)=\sup_{s+r=t, s,r\in R}\Delta (F_{x,z}(s),F_{y,z}(r)),\quad t\in R. \] By virtue of this definition, some convergence theorems, Schwarz inequality and the orthogonal concept for probabilistic inner product spaces are established and introduced. Moreover, the relationship between this kind of spaces and probabilistic normed spaces is considered also.
scientific article; zbMATH DE number 3996388