scholarly journals Angle in the space of $ p $-summable sequences

2021 ◽  
Vol 7 (2) ◽  
pp. 2810-2819
Author(s):  
Muh Nur ◽  
◽  
Moch Idris ◽  
Firman ◽  
Keyword(s):  
Dimensional Subspace ◽  
Inner Product ◽  
Usual Norm

<abstract><p>The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1\le p &lt; \infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ s\ge 2 $ of $ A $.</p></abstract>

scholarly journals Characterizations of Operator Birkhoff–James Orthogonality

2017 ◽  
Vol 60 (4) ◽  
pp. 816-829 ◽  
Author(s):  
Mohammad Sal Moslehian ◽  
Ali Zamani
Keyword(s):  
Unit Vector ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Inner Product ◽  
Finite Dimensional Subspace ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
James Orthogonality ◽  
New Type ◽  
Norm Attaining

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.

scholarly journals Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 941 ◽  
Author(s):  
Milton Ferreira ◽  
Teerapong Suksumran
Keyword(s):  
General Theory ◽  
Decomposition Theorem ◽  
Dimensional Subspace ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Fiber Bundles ◽  
Finite Dimensional Subspace ◽  
Isomorphism Theorem ◽  
Quotient Spaces ◽  
Finite Dimensional

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.

Geodesic vectors and subalgebras in two-step nilpotent metric Lie algebras

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Péter T. Nagy ◽  
Szilvia Homolya
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Structure ◽  
Dimensional Subspace ◽  
Inner Product ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Left Invariant Riemannian Metric ◽  
Zero Vector

AbstractA metric Lie algebra is a Lie algebra endowed with a Euclidean inner product. A subalgebra is called flat, respectively totally geodesic, if its exponential image in the corresponding Lie group with left invariant Riemannian metric is flat, respectively a totally geodesic submanifold. A non-zero vector is geodesic, if the generated one-dimensional subspace is totally geodesic. We study geodesic vectors and flat totally geodesic subalgebras in two-step nilpotent metric Lie algebras and show that their linear structure is independent of the inner product of the metric Lie algebra. We determine the geodesic vectors and the flat totally geodesic subalgebras in the two-step nilpotent metric Lie algebras of dimension ≤ 6.

scholarly journals TEOREMA TITIK TETAP PADA RUANG NORM-n STANDAR

2012 ◽  
Vol 4 (1) ◽  
pp. 69
Author(s):  
Shelvi Ekariani ◽  
Hendra Gunawan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Usual Norm

On the standard n-normed space,i.e an inner product space equipped with the standard n-norm, one can derive a norm from the n-norm in a certain way. The purpose of this note is to establish the equivalence between such a norm and the usual norm on standard n-normed space. Further, this fact together with others use to prove  a fixed point theorem on the standard n-normed space. 

scholarly journals Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups

2020 ◽  
Author(s):  
Milton Ferreira ◽  
Teerapong Suksumran
Keyword(s):  
General Theory ◽  
Decomposition Theorem ◽  
Dimensional Subspace ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Fiber Bundles ◽  
Finite Dimensional Subspace ◽  
Isomorphism Theorem ◽  
Quotient Spaces ◽  
Finite Dimensional

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, M\"{o}bius, Proper Velocity, and Chen's gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite-dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.

Inner Product Performance Criteria for Evaluating Combat Models

1991 ◽  
Author(s):  
Herbert E. Cohen
Keyword(s):  
Product Performance ◽  
Performance Criteria ◽  
Inner Product

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete 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.

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