It was also found that the intermodule synchronization was more noticeable when the inner-module synchronization was weak.
All the basic notions such as random normed modules, random inner product modules and random locally convex modules, together with their random conjugate spaces, were naturally presented by Guo in the course of the development of random functional analysis [1 4].
Guo initiated a new approach to random functional analysis [1 3], whose main idea is to develop random functional analysis as functional analysis based on random normed modules, random inner product modules and random locally convex modules.
If, in addition, (v) implies, then is called an inner product module over.
As we can see, an inner product module obeys the same axioms as an ordinary inner product space, except that the inner product takes values in a more general structure than in the field of complex numbers.
A semi-inner product module over is a right module over together with a generalized semi-inner product, that is with a mapping on, which is -valued if is a proper -algebra, or -valued if is a -algebra, having the following properties: (i) for all, (ii) for,, (iii) for all, (iv) for.
A semi-inner product module over A is a right module X over A together with a generalized semi-inner product, that is, with a mapping 〈 ⋅, ⋅ 〉 on X × X, which is A -valued and has the following properties: (i) 〈 x, y + z 〉 = 〈 x, y 〉 + 〈 x, z 〉 for all x, y, z ∈ X, (ii) 〈 x, y a 〉 = 〈 x, y 〉 a for x, y ∈ X, a ∈ A, (iii) 〈 x, y 〉 ∗ = 〈 y, x 〉 for all x, y ∈ X, (iv) 〈 x, x 〉 ≥ 0 for x ∈ X. .
On the roof baffle and on the inner baffle modules, no clear reduction of the film thickness was found.
The Grüss inequality has been investigated in inner product modules over H ∗ -algebras and C ∗ -algebras [4, 5], completely bounded maps [6], n-positive linear maps [7] and semi-inner product C ∗ -modules [8].
We obtain some further generalization of the Grüss type inequalities in inner product modules over proper -algebras and unital Banach -algebras for -seminorms and positive linear functionals.
