scholarly journals Operator inequalities related to p-angular distances

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2107-2111
Author(s):  
Taba Afkhami ◽  
Hossein Dehghan
Keyword(s):  
Normed Space ◽  
Angular Distance ◽  
Inner Product ◽  
Product Spaces ◽  
Operator Inequalities ◽  
Inner Product Spaces ◽  
Operator Version

For any nonzero elements x,y in a normed space X, the angular and skew-angular distance is respectively defined by ?[x,y] = ||x/||x|| - y/||y|||| and ?[x,y] = ||x/||y|| - y/||x||||. Also inequality ? ? ? characterizes inner product spaces. Operator version of ? p has been studied by Pecaric, Rajic, and Saito, Tominaga, and Zou et al. In this paper, we study the operator version of p-angular distance ?p by using Douglas? lemma. We also prove that the operator version of inequality ? p ? ?p holds for normal and double commute operators. Some examples are presented to show essentiality of these conditions.

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.

On a Characterisation of Inner Product Spaces

2001 ◽  
Vol 8 (2) ◽  
pp. 231-236
Author(s):  
G. Chelidze
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Normed Space ◽  
Inner Product ◽  
Product Spaces ◽  
Second Moment ◽  
Inner Product Spaces ◽  
Minimum Value ◽  
Positive Weights ◽  
The Mean

Abstract It is well known that for the Hilbert space H the minimum value of the functional F μ (f) = ∫ H ‖f – g‖2 dμ(g), f ∈ H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

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.

Quasi-inner product spaces of quasi-Sobolev spaces and their completeness

2018 ◽  
pp. 339
Author(s):  
Jawad Kadhim Khalaf Al-Delfi
Keyword(s):  
Hilbert Space ◽  
Sobolev Spaces ◽  
Hilbert Spaces ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

      Sequences spaces  , m  ,  p  have called quasi-Sobolev spaces were  introduced   by Jawad . K. Al-Delfi in 2013  [1]. In this  paper , we deal with notion of  quasi-inner product  space  by using concept of  quasi-normed  space which is generalized  to normed space and given a  relationship  between  pre-Hilbert space and a  quasi-inner product space with important  results   and   examples.  Completeness properties in quasi-inner   product space gives  us  concept of  quasi-Hilbert space .  We show  that ,  not  all  quasi-Sobolev spaces  ,  are  quasi-Hilbert spaces. The  best  examples which are  quasi-Hilbert spaces and Hilbert spaces  are , where  m  . Finally, propositions, theorems and examples are our own unless otherwise referred.    

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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