scholarly journals Some geometric characterizations of inear product spaces

1981 ◽  
Vol 24 (2) ◽  
pp. 239-246 ◽  
Author(s):  
O. P. Kapoor ◽  
S. B. Mathur
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Normed Linear Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Linear Spaces ◽  
Von Neumann ◽  
James Orthogonality

There are several geometric characterizations of inner product spaces amongst the normed linear spaces. Mahlon M. Day's refinement “rhombi suffice as well as parallelograms”, of P. Jordan and J. von Neumann parallelogram law is well known. There are some characterizations which employ various notions of orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality. For example, it is known that if in a normed linear space Birkhoff-James orthogonality implies isosceles orthogonality then the space is an inner product space; geometrically it means that if the diagonals of a rectangle, with sides perpendicular in Birkhoff-James sense, are equal then the space is an inner product space. In the main result of this note we improve upon this characterization and show that here unit squares suffice as well as rectangles.

scholarly journals Orthogonality and characterizations of inner product spaces

1978 ◽  
Vol 19 (3) ◽  
pp. 403-416 ◽  
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.

scholarly journals On Carlsson type orthogonality and characterization of inner product spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 859-870 ◽  
Author(s):  
Eder Kikianty ◽  
Sever Dragomir
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Inner Product Spaces ◽  
Pythagorean Orthogonality

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.

scholarly journals Some Properties of Fuzzy Anti-Inner Product Spaces

2020 ◽  
pp. 3042-3047
Author(s):  
Radhi I. M. Ali ◽  
Esraa A. Hussein
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Definition Of

In this paper, the definition of fuzzy anti-inner product in a linear space is introduced. Some results of fuzzy anti-inner product spaces are given, such as the relation between fuzzy inner product space and fuzzy anti-inner product. The notion of minimizing vector is introduced in fuzzy anti-inner product settings.

CHEBYSHEV SETS

2014 ◽  
Vol 98 (2) ◽  
pp. 161-231 ◽  
Author(s):  
JAMES FLETCHER ◽  
WARREN B. MOORS
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Normed Linear Space ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Inner Product Space ◽  
Inner Product ◽  
Best Approximations ◽  
Chebyshev Set ◽  
Chebyshev Sets

AbstractA Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

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.

scholarly journals On Bessel and Grüss inequalities for orthonormal families in inner product spaces

2004 ◽  
Vol 69 (2) ◽  
pp. 327-340 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

A new reverse of Bessel's inequality for orthornormal families in real or complex inner product space is obtained. Applications for some Grüss type results are also provided.

scholarly journals Characterizations of inner product spaces by means of norm one points

2005 ◽  
Vol 97 (1) ◽  
pp. 104
Author(s):  
José Mendoza ◽  
Tijani Pakhrou
Keyword(s):  
Convex Hull ◽  
Unit Sphere ◽  
Product Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Chebyshev Center ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Point Subset ◽  
Real Normed Linear Space

Let $X$ be a a real normed linear space of dimension at least three, with unit sphere $S_X$. In this paper we prove that $X$ is an inner product space if and only if every three point subset of $S_X$ has a Chebyshev center in its convex hull. We also give other characterizations expressed in terms of centers of three point subsets of $S_X$ only. We use in these characterizations Chebyshev centers as well as Fermat centers and $p$-centers.

scholarly journals On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajracharya ◽  
Vishnu Narayan Mishra
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Isosceles Orthogonality ◽  
Complete Proof ◽  
Underlying Space ◽  
Real Inner Product Space ◽  
Special Case ◽  
Real Normed Linear Space ◽  
Pythagorean Orthogonality

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

A proof of Garfunkel inequality and of some related results in inner product spaces

2017 ◽  
Vol 26 (2) ◽  
pp. 153-162
Author(s):  
DAN S¸ TEFAN MARINESCU ◽  
MIHAI MONEA
Keyword(s):  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Geometric Inequality ◽  
Product Spaces ◽  
Inner Product Spaces

In this paper, we will present a inner product space proof of a geometric inequality proposed by J. Garfunkel in American Mathematical Monthly [Garfunkel, J., Problem 2505, American Mathematical Monthly, 81 (1974), No. 11] and consider some other similar results.

scholarly journals Norms on Quotient Spaces of The 2-Inner Product Space

2021 ◽  
Vol 3 (2) ◽  
pp. 80
Author(s):  
Harmanus Batkunde
Keyword(s):  
Product Space ◽  
Independent Set ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Inner Products

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.

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