Characterization Of Inner Product Spaces Research Articles

Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces

Abstract This article presents the general solution f : G → V f:{\mathcal{G}}\to {\mathcal{V}} of the following functional equation: f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G , f\left(x)-4f\left(x+y)+6f\left(x+2y)-4f\left(x+3y)+f\left(x+4y)=0,\hspace{1.0em}x,y\in {\mathcal{G}}, where ( G , + ) \left({\mathcal{G}},+) is an abelian group and V {\mathcal{V}} is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.

The Best Ulam Constant of the Fréchet Functional Equation

In this paper, we prove the Ulam stability of the Fréchet functional equation f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(z+x) arising from the characterization of inner product spaces and we determine its best Ulam constant. Using this result, we give a stability result for a pexiderized version of the Fréchet functional equation.

Characterization of inner product spaces by unitary Carlsson type orthogonalities

In this study, we consider the Hermite-Hadamard type of unitary Carlsson?s orthogonality (UHH-C-orthogonality) to characterize real inner product spaces. We give a necessary and sufficient condition weaker than the homogeneity of symmetric HH-C-orthogonalities which characterizes inner product spaces among normed linear spaces of dimension at least three. In conclusion, some more characterizations of real inner product spaces are provided.

Additive and Fréchet functional equations on restricted domains with some characterizations of inner product spaces

<abstract><p>In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.</p></abstract>

Homogeneity of isosceles orthogonality, transitivity of the norm, and characterizations of inner product spaces

Homogeneity of isosceles orthogonality, transitivity of the norm, and characterizations of inner product spaces

Bounds for the p-Angular Distance and Characterizations of Inner Product Spaces

Based on a suitable improvement of a triangle inequality, we derive new mutual bounds for p-angular distance \(\alpha _p[x,y]=\big \Vert \Vert x\Vert ^{p-1}x- \Vert y\Vert ^{p-1}y\big \Vert \), in a normed linear space X. We show that our estimates are more accurate than the previously known upper bounds established by Dragomir, Hile and Maligranda. Next, we give several characterizations of inner product spaces with regard to the p-angular distance. In particular, we prove that if \(|p|\ge |q|\), \(p\ne q\), then X is an inner product space if and only if for every \(x,y\in X{\setminus } \{0\}\), $$\begin{aligned} {\alpha _p[x,y]}\ge \frac{{\Vert x\Vert ^{p}+\Vert y\Vert ^{p} }}{\Vert x\Vert ^{q}+\Vert y\Vert ^{q} }\alpha _q[x,y]. \end{aligned}$$

Inner product spaces and quadratic functional equations

In this paper, we introduce the functional equations f(2x−y)+f(x+2y)=5[f(x)+f(y)],f(2x−y)+f(x+2y)=5f(x)+4f(y)+f(−y),f(2x−y)+f(x+2y)=5f(x)+f(2y)+f(−y),f(2x−y)+f(x+2y)=4[f(x)+f(y)]+[f(−x)+f(−y)].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} f(2x-y)+f(x+2y)&=5\\bigl[f(x)+f(y)\\bigr], \\\\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=4\\bigl[f(x)+f(y)\\bigr]+\\bigl[f(-x)+f(-y)\\bigr]. \\end{aligned}$$ \\end{document} We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

Characterizations of inner product spaces via angular distances and Cauchy–Schwarz inequality

We study some interesting characterizations of real inner product spaces expressed in terms of angular distances. We first discuss the equivalence of characterizing an inner product space via the usual angular distance and the p-angular distance. Then, we establish a parametric family of upper bounds for the usual angular distance which also serves as a characterization of an inner product space. As an application, bounds for the usual angular distance are utilized in obtaining improvements of the real Cauchy–Schwarz inequality. Finally, we give several comparative relations for angular distances in inner product spaces.

Characterization of Inner Product Spaces by Strongly Schur-Convex Functions

Involving the notion of strongly Schur-convex functions we give a new characterization of inner product spaces among norm spaces. We also present a representation theorem for functions which generate strongly Schur-convex sums.

Stability of the Fréchet Equation in Quasi-Banach Spaces

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-Banach spaces. In the proof, a fixed point theorem due to Dung and Hang, which is an extension of a fixed point theorem in Banach spaces, plays a main role.

Characterization of inner product spaces and quadratic functions by some classes of functions

Characterization of inner product spaces and quadratic functions by some classes of functions

Extension of Dunkl--Williams inequality and characterizations of inner product spaces

Given $ p, q \in \mathbb {R} $, we generalize the classical Dunkl--Williams inequality for $p$-angular and $q$-angular distances in inner product spaces. We extend the Hile inequality for arbitrary $p$-angular and $q$-angular distances and study some geometric aspects of a generalization of Dunkl--Williams inequality. Introducing power refinements, we show significant power refinements of the generalized Dunkl--Williams inequality under some mild conditions. Among other things, we give new characterizations of inner product spaces with regard to $p$-angular and $q$-angular distances. In particular, we prove that if $ p, q, r \in \mathbb {R} $, $ q \neq 0$ and $ 0\leq p/q\lt 1 $, then $ X $ is an inner product space if and only if for every $ x, y \in X \setminus \{0\} $, $$ \bigl \lVert \lVert x \rVert ^{p-1} x - \lVert y \rVert ^{p-1}y \bigr \rVert \leq \frac {2^{1/r}\bigl \lVert \lVert x \rVert ^{q-1} x - \lVert y \rVert ^{q-1}y \bigr \rVert }{\bigl [\lVert x \rVert ^{r(q-p)} + \lVert y \rVert ^{r(q-p)}\bigr ]^{\frac {1}{r}}}. $$

Some characterizations of inner product spaces based on angle

A problem in functional analysis that arises naturally is about finding necessary and sufficient conditions for a normed space to be an inner product space. By answering this question, mathematicians try to understand the inner product and normed spaces features. In this note, we have discussed this issue and we prove some results concerned with it. We introduce a notion of angle between two vectors in a normed space, denoted by $A_\theta(.,.)$ where $\theta\neq{k\pi\over2}$. We also speak about a notion of orthogonality concerning it, we call it $\theta$-orthogonality.

Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces

Abstract We have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

An extension of orthogonality relations based on norm derivatives

Abstract We introduce the relation ρλ-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional ρλ. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving ρλ-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.

Geometric Aspects of $p$-angular and Skew $p$-angular Distances

Corresponding to the concept of $p$-angular distance $\alpha_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-angular distance $\beta_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x, y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances. We then give some results comparing two different $p$-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $\alpha_p[x,y]\geq\beta_p[x,y]$.

On the generalized Fr\xe9chet functional equation with constant coefficients and its stability

We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to show the existence of an exact solution of the equation that is close to a given approximate solution.

Norm inequalities and characterizations of inner product spaces

Norm inequalities and characterizations of inner product spaces

Norm inequalities and characterizations of inner product spaces

Norm inequalities and characterizations of inner product spaces

A support theorem for generalized convexity and its applications

In the present paper we introduce a notion of (ω,t)-convexity as a natural generalization of the notion of usual t-convexity, t-strongly convexity, approximate t-convexity, delta t-convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an (ω,t)-convex map can be supported at a given point by an (ω,t)-affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of (ω,t)-convexity are given.