Weak additivity of metric pythagorean orthogonality

1995 ◽  
Vol 54 (1-2) ◽  
pp. 44-49 ◽  
Author(s):  
Raymond Freese ◽  
Edward Andalafte
Keyword(s):  
Pythagorean Orthogonality

Pythagorean orthogonality of compact sets

2018 ◽  
Vol 11 (5) ◽  
pp. 735-752
Author(s):  
Pallavi Aggarwal ◽  
Steven Schlicker ◽  
Ryan Swartzentruber
Keyword(s):  
Compact Sets ◽  
Pythagorean Orthogonality

scholarly journals Circle-Uniqueness of Pythagorean Orthogonality in Normed Linear Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Senlin Wu ◽  
Xinjian Dong ◽  
Dan Wang
Keyword(s):  
Unit Sphere ◽  
Linear Spaces ◽  
Strictly Convex ◽  
Normed Linear Spaces ◽  
Underlying Space ◽  
Pythagorean Orthogonality

We introduce the circle-uniqueness of Pythagorean orthogonality in normed linear spaces and show that Pythagorean orthogonality is circle-unique if and only if the underlying space is strictly convex. Further related results providing more detailed relations between circle-uniqueness of Pythagorean orthogonality and the shape of the unit sphere are also presented.

scholarly journals Pythagorean Orthogonality in a Normed Linear Space

1958 ◽  
Vol 9 (4) ◽  
pp. 168-169
Author(s):  
Hazel Perfect
Keyword(s):  
Euclidean Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Image Position ◽  
Pythagorean Orthogonality

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

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 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.

Pythagorean orthogonality and additive mappings

1997 ◽  
Vol 53 (1-2) ◽  
pp. 108-126 ◽  
Author(s):  
György Szabó
Keyword(s):  
Additive Mappings ◽  
Pythagorean Orthogonality

scholarly journals Some New Types of Orthogonalities in Normed Spaces and Application in Best Approximation

2016 ◽  
Vol 2 ◽  
pp. 1
Author(s):  
Bhuwan Prasad Ojha
Keyword(s):  
Banach Spaces ◽  
Best Approximation ◽  
Birkhoff Orthogonality ◽  
Normed Spaces ◽  
College Of Engineering ◽  
Pythagorean Orthogonality

<p>In this paper, two new types of orthogonality from the generalized Carlssion orthogonality have been studied and some properties of orthogonality in Banach spaces are verified and as best implies Birkhoff orthogonality and Birkhoff orthogonality implies best approximation, in this paper, Pythagorean orthogonality also implies best approximation has been proved.</p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol. 2, 2016, page: 1-4</p>

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 Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space

2019 ◽  
Vol 4 (4) ◽  
pp. 72-78
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajrayacharya
Keyword(s):  
Normed Space ◽  
Best Approximation ◽  
Normed Linear Space ◽  
Inner Product ◽  
Birkhoff Orthogonality ◽  
Isosceles Orthogonality ◽  
Best Approximations ◽  
Linear Spaces ◽  
The Best Approximation ◽  
Pythagorean Orthogonality

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

Weak homogeneity of metric pythagorean orthogonality

1996 ◽  
Vol 56 (1-2) ◽  
pp. 3-8 ◽  
Author(s):  
Edward Andalafte ◽  
Raymond Freese
Keyword(s):  
Weak Homogeneity ◽  
Pythagorean Orthogonality
Sign in / Sign up

Export Citation Format

Share Document