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.

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 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.

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 Finite Dimensional Chebyshev Subspaces of Lo

2017 ◽  
Vol 22 (1) ◽  
pp. 53
Author(s):  
Aref K. Kamal
Keyword(s):  
Linear Space ◽  
Best Approximation ◽  
Normed Linear Space ◽  
The Best Approximation ◽  
Finite Dimensional

If A is a subset of the normed linear space X, then A is said to be proximinal in X if for each xÎX there is a point y0ÎA such that the distance between x and A; d(x, A) = inf{||x-y||: yÎA}= ||x­-y0||. The element y0 is called a best approximation for x from A. If for each xÎX, the best approximation for x from A is unique then the subset A is called a Chebyshev subset of X.  In this paper the author studies the existence of finite dimensional Chebyshev subspaces of Lo.   

scholarly journals Aplikasi Pemetaan terkait Semi-Inner Produk pada Ruang Bernorma Real

2021 ◽  
Vol 1 (1) ◽  
pp. 7-14
Author(s):  
Aulia Khifah Futhona ◽  
Supama
Keyword(s):  
Normed Space ◽  
Inner Product ◽  
Birkhoff Orthogonality ◽  
Real Normed Space ◽  
Lower Semi

In this article, we give the properties of mappings associated with the upper semi-inner product , lower semi-inner product  and Lumer semi-inner product  which generate the norm on a real normed space. Furthermore, we establish applications to the Birkhoff orthogonality and characterization of best approximants.

scholarly journals Fuzzyn-normed linear space

2005 ◽  
Vol 2005 (24) ◽  
pp. 3963-3977 ◽  
Author(s):  
AL. Narayanan ◽  
S. Vijayabalaji
Keyword(s):  
Linear Space ◽  
Normed Space ◽  
Best Approximation ◽  
Normed Linear Space

The primary purpose of this paper is to introduce the notion of fuzzyn-normed linear space as a generalization ofn-normed space. Ascending family ofα-n-norms corresponding to fuzzyn-norm is introduced. Best approximation sets inα-n-norms are defined. We also provide some results on best approximation sets inα-n-normed space.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

2008 ◽  
Vol 06 (02) ◽  
pp. 315-330 ◽  
Author(s):  
PETER BALAZS
Keyword(s):  
Hilbert Spaces ◽  
Best Approximation ◽  
Inner Product ◽  
Riesz Bases ◽  
Wavelet Frames ◽  
Arbitrary Matrix ◽  
The Best Approximation ◽  
Rank One ◽  
Arbitrary Operator ◽  
Bessel Sequences

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

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>

Sign in / Sign up

Export Citation Format

Share Document