Abstract
In this article, we established new results related to a 2-pre-Hilbert space. Among these results we will mention the Cauchy-Schwarz inequality. We show several applications related to some statistical indicators as average, variance and standard deviation and correlation coefficient, using the standard 2-inner product and some of its properties. We also present a brief characterization of a linear regression model for the random variables in discrete case.
Highlights
In Reference [1] Gähler introduced the definitions of a linear 2-normed space and a 2-metric space.In References [2,3], Diminnie, Gähler and White studied the properties of a 2-inner product space.Several results related to the theory of 2-inner product spaces can be found in Reference [4].In Reference [5] Dragomir et al show the corresponding version of Boas-Bellman inequality in 2-inner product spaces and in Reference [6] the superadditivity and the monotony of 2-norms generated by inner products was studied.We consider X a linear space of dimension greater than 1 over the field K, where K is the set of the real or the complex numbers
A reverse of the Cauchy-Schwarz inequality in 2-inner product spaces can be found in Reference [5]: if u, v, w ∈ X and a, A ∈ K are such that Re( Av − u, u − av|w) ≥ 0 or equivalently u−
In Reference [12], we find some refinements of Ostrowski’s inequality and an extention to a 2-inner product space
Summary
In Reference [1] Gähler introduced the definitions of a linear 2-normed space and a 2-metric space. In References [2,3], Diminnie, Gähler and White studied the properties of a 2-inner product space. A function k · | · k defined on X × X and satisfying the above conditions is called 2-norm on X and ( X, k · | · k) is called linear 2-normed space. A reverse of the Cauchy-Schwarz inequality in 2-inner product spaces can be found in Reference [5]: if u, v, w ∈ X and a, A ∈ K are such that Re( Av − u, u − av|w) ≥ 0 or equivalently u−. The Cauchy-Schwarz inequality in the real case, |hu, vi| ≤ kuk · kvk (see e.g., References [9,10]), can be obtained by the following identity, as in Reference [11],. We present a brief characterization of a linear regression model for the random variables in discrete case
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