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 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 The constants to measure the differences between Birkhoff and isosceles orthogonalities

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2761-2770 ◽  
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).

scholarly journals Characterizations of inner product spaces by orthogonal vectors

2006 ◽  
Vol 4 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Marco Baronti ◽  
Emanuele Casini
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Closed Ball ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Real Normed Space

LetXbe a real normed space with unit closed ballB. We prove thatXis an inner product space if and only if it is true that wheneverx,yare points in?Bsuch that the line throughxandysupports22Bthenx?yin the sense of Birkhoff.

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 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 Some properties of the superior and inferior semi inner product function associated to the 2-norm

2016 ◽  
Vol 12 (5) ◽  
pp. 6254-6260
Author(s):  
Stela Ceno
Keyword(s):  
Normed Space ◽  
Scalar Product ◽  
Inner Product ◽  
Normed Spaces ◽  
Close Link ◽  
Long Period ◽  
Product Function

Special properties that the scalar product enjoys and its close link with the norm function have raised the interest of researchers from a very long period of time. S.S. Dragomir  presents concrete generalizations of the scalar product functions  in a normed space and deals with the interesting properties of them. Based on S.S. Dragomirs idea in this paper we treat generalizations of superior and inferior scalar product functions in the case of semi-normed spaces and 2-normed spaces.

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

scholarly journals Some Properties oflp(A,X)Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaohong Fu ◽  
Songxiao Li
Keyword(s):  
Simple Proof ◽  
Normed Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Locally Convex Space ◽  
Locally Convex

We provide a representation of elements of the spacelp(A,X)for a locally convex spaceXand1≤p<∞and determine its continuous dual for normed spaceXand1<p<∞. In particular, we study the extension and characterization of isometries onlp(N,X)space, whenXis a normed space with an unconditional basis and with a symmetric norm. In addition, we give a simple proof of the main result of G. Ding (2002).

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