scholarly journals Some equivalent characterizations of inner product spaces and their consequences

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1587-1599 ◽  
Author(s):  
Dan Marinescu ◽  
Mihai Monea ◽  
Mihai Opincariu ◽  
Marian Stroe
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces

In this paper, first we prove the equivalence of the Frechet and the Jordan-Neumann relations in normed spaces. Next, we will present some characterizations of the inner product spaces and we will show their equivalence. Finally, we will prove some consequences of our main results.

scholarly journals The constants to measure the differences between Birkhoff and isosceles orthogonalities

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2761-2770 ◽  
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).

scholarly journals Inner product spaces and quadratic functional equations

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.

scholarly journals Intuitionistic Fuzzy Stability of Functional Equations Associated with Inner Product Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Zhihua Wang ◽  
Themistocles M. Rassias
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces ◽  
Intuitionistic Fuzzy ◽  
Stability Of Functional Equations ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Stability Results

In intuitionistic fuzzy normed spaces, we investigate some stability results for the functional equation which is said to be a functional equation associated with inner products space.

scholarly journals An extension of orthogonality relations based on norm derivatives

2018 ◽  
Vol 70 (2) ◽  
pp. 379-393 ◽  
Author(s):  
Ali Zamani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Linear Mapping ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces ◽  
Scalar Multiple ◽  
Essential Properties ◽  
Orthogonality Relations ◽  
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.

scholarly journals A Functional equation related to inner product spaces in non-archimedean normed spaces

2011 ◽  
Vol 2011 (1) ◽  
pp. 37
Author(s):  
Madjid Gordji ◽  
Razieh Khodabakhsh ◽  
Hamid Khodaei ◽  
Choonkil Park ◽  
Dong shin
Keyword(s):  
Functional Equation ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

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