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.

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The constants to measure the differences between Birkhoff and isosceles orthogonalities

Filomat ◽

10.2298/fil1610761m ◽

2016 ◽

Vol 30 (10) ◽

pp. 2761-2770 ◽

Cited By ~ 2

Author(s):

Hiroyasu Mizuguchi

Keyword(s):

Normed Space ◽

Inner Product ◽

Product Spaces ◽

Normed Spaces ◽

Isosceles Orthogonality ◽

Quantitative Characterization ◽

Inner Product Spaces ◽

The Difference ◽

Geometric Constant

The notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful. When moving to normed spaces, we have many possibilities to extend this notion. We consider Birkhoff orthogonality and isosceles orthogonality, which are the most used notions of orthogonality. In 2006, Ji and Wu introduced a geometric constant D(X) to give a quantitative characterization of the difference between these two orthogonality types. However, this constant was considered only in the unit sphere SX of the normed space X. In this paper, we introduce a new geometric constant IB(X) to measure the difference between Birkhoff and isosceles orthogonalities in the entire normed space X. To consider the difference between these orthogonalities, we also treat constant BI(X).

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Inner product spaces and quadratic functional equations

Advances in Difference Equations ◽

10.1186/s13662-021-03307-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Jae-Hyeong Bae ◽

Batool Noori ◽

M. B. Moghimi ◽

Abbas Najati

Keyword(s):

Functional Equations ◽

Quadratic Functions ◽

Inner Product ◽

Quadratic Functional ◽

Product Spaces ◽

Normed Spaces ◽

Inner Product Spaces ◽

Restricted Domains ◽

The Stability

AbstractIn this paper, we introduce the functional equations $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ f ( 2 x − y ) + f ( x + 2 y ) = 5 [ f ( x ) + f ( y ) ] , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + 4 f ( y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + f ( 2 y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 4 [ f ( x ) + f ( y ) ] + [ f ( − x ) + f ( − y ) ] . 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.

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