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

Author(s):  
Harmanus Batkunde

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.

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida

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.


2004 ◽  
Vol 69 (2) ◽  
pp. 327-340 ◽  
Author(s):  
S. S. Dragomir

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.


Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 859-870 ◽  
Author(s):  
Eder Kikianty ◽  
Sever Dragomir

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.


2017 ◽  
Vol 26 (2) ◽  
pp. 153-162
Author(s):  
DAN S¸ TEFAN MARINESCU ◽  
MIHAI MONEA

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.


2020 ◽  
pp. 3042-3047
Author(s):  
Radhi I. M. Ali ◽  
Esraa A. Hussein

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.


2005 ◽  
Vol 2005 (18) ◽  
pp. 2883-2893 ◽  
Author(s):  
A. H. Ansari ◽  
M. S. Moslehian

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that ifais a unit vector in a real or complex inner product space(H;〈.,.〉),r,s>0,p∈(0,s],D={x∈H,‖rx−sa‖≤p},x1,x2∈D−{0}, andαr,s=min{(r2‖xk‖2−p2+s2)/2rs‖xk‖:1≤k≤2}, then(‖x1‖‖x2‖−Re〈x1,x2〉)/(‖x1‖+‖x2‖)2≤αr,s.


2005 ◽  
Vol 78 (2) ◽  
pp. 199-210 ◽  
Author(s):  
Emmanuel Chetcuti ◽  
Anatolij Dvurečenskij

AbstractWe introduce sign-preserving charges on the system of all orthogonally closed subspaces, F(S), of an inner product space S, and we show that it is always bounded on all the finite-dimensional subspaces whenever dim S = ∞. When S is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that S is complete whenever F(S) admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.


Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 571
Author(s):  
Taechang Byun ◽  
Ji Eun Lee ◽  
Keun Young Lee ◽  
Jin Hee Yoon

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved.


2020 ◽  
Vol 57 (4) ◽  
pp. 541-551
Author(s):  
Behrooz Mohebbi Najmabadi ◽  
Tayebe Lal Shateri ◽  
Ghadir Sadeghi

In this paper, we define an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2-*-inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an A-2-inner product space.


2006 ◽  
Vol 4 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Marco Baronti ◽  
Emanuele Casini

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.


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