C. Dumitrescu gave a definition of probabilistic inner product spaces in 1981. The authors of this paper modified his definition as follows. (E,F,T) is called a PIP space, if E is a linear space; T is a weak t- norm; \({\mathcal F}\) is the set of all left-continuous distribution functions; F is a map from \(E\times E\) to \({\mathcal F}\) with the following properties:
(1) \(F_{(x,x)}(0)=0\); \(F_{(x,x)}(t)=H(t)\) (Heaviside function) \(\Leftrightarrow\) \(x=0\); (2) \(F_{(x,y)}=F_{(y,x)}\); (3) \(F_{(\lambda x,y)}(t)=F_{(x,y)}(\frac{t}{\lambda})\), \(\forall \lambda >0\), \(F_{(0x,y)}(t)=H(t)\); \(F_{(-x,y)}(t)=1-\lim_{t'\to t^- }F_{(x,y)}(-t')\); (4) \(S(F_{(x,z)}(t_ 1),F_{(y,z)}(t_ 2))\geq F_{(x+y,z)}(t_ 1+t_ 2)\geq T(F_{(x,z)}(t_ 1),F_{(y,2)}(t_ 2))\), \(\forall t_ 1,t_ 2\in R\), where \(S(a,b)=1-T(1-a,1-b).\)
The main results are the following.
Theorem 2. If (E,F,T) is a PIP space and \(\sup_{a<1}T(a,a)=1\), then there is a sufficient family of pseudo inner products \(\{(\cdot,\cdot)_ r\), \(r\in (0,1)\}.\)
Theorem 5. (E,F,T) isometrically isomorphic to an inner product space iff \(T=\min.\)
[For part I see ibid. 17, No.3, 123-125 (1983; Zbl 0536.54005).]