On the probabilistic inner product spaces
type: Item
instance of: scholarly article
title: On the probabilistic inner product spaces
zbMATH Open document ID: 0615.46024
DOI: 10.1007/BF01897206
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 54A99
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 3996388
zbMATH Keywords: probabilistic inner product space
zbMATH Keywords: Schwarz inequality
zbMATH Keywords: orthogonal concept
author: Shihsen Chang
MaRDI profile type: Publication
cites work: Q3336431
cites work: Statistical metric spaces
cites work: Q3914130
description: scientific article; zbMATH DE number 3996388