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

In an inner product space, two vectors are orthogonal if their inner product is zero. In a normed space, numerous notions of orthogonality have been introduced via equivalent propositions to the usual orthogonality, e.g. orthogonal vectors satisfy the Pythagorean law. In 2010, Kikianty and Dragomir [9] introduced the p-HH-norms (1 ? p < ?) on the Cartesian square of a normed space. Some notions of orthogonality have been introduced by utilizing the 2-HH-norm [10]. These notions of orthogonality are closely related to the classical Pythagorean orthogonality and Isosceles orthogonality. In this paper, a Carlsson type orthogonality in terms of the 2-HH-norm is considered, which generalizes the previous definitions. The main properties of this orthogonality are studied and some useful consequences are obtained. These consequences include characterizations of inner product space.

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An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

Journal of Approximation Theory ◽

10.1016/0021-9045(83)90073-4 ◽

1983 ◽

Vol 39 (4) ◽

pp. 295-298 ◽

Cited By ~ 6

Author(s):

Charles R Diminnie ◽

Raymond W Freese ◽

Edward Z Andalafte

Keyword(s):

Inner Product ◽

Product Spaces ◽

Isosceles Orthogonality ◽

Inner Product Spaces

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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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Orthogonality and characterizations of inner product spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700008947 ◽

1978 ◽

Vol 19 (3) ◽

pp. 403-416 ◽

Cited By ~ 13

Author(s):

O.P. Kapoor ◽

Jagadish Prasad

Keyword(s):

Linear Space ◽

Normed Linear Space ◽

Inner Product ◽

Product Spaces ◽

Isosceles Orthogonality ◽

Linear Spaces ◽

Strictly Convex ◽

Inner Product Spaces ◽

James Orthogonality ◽

Pythagorean Orthogonality

Using the notions of orthogonality in normed linear spaces such as isosceles, pythagorean, and Birkhoff-James orthogonality, in this paper we provide some new characterizations of inner product spaces besides giving simpler proofs of existing similar characterizations. In addition we prove that in a normed linear space pythagorean orthogonality is unique and that isosceles orthogonality is unique if and only if the space is strictly convex.

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Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽

10.3390/math9070765 ◽

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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