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 New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 951
Author(s):  
Yuankang Fu ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Upper And Lower Bounds ◽  
Uniformly Convex ◽  
Equilateral Triangles ◽  
Geometric Constant ◽  
Geometric Constants

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

scholarly journals Some aspects of generalized Zbăganu and James constant in Banach spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 299-310
Author(s):  
Qi Liu ◽  
Muhammad Sarfraz ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Point Property ◽  
James Constant ◽  
Geometric Constant

Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

scholarly journals Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1294
Author(s):  
Asif Ahmad ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Basic Properties ◽  
Geometric Constant ◽  
Geometric Constants

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

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).

Diamonds II

2020 ◽  
pp. 64-73
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
The Other ◽  
Equivalent Characterization ◽  
Perfectoid Spaces ◽  
Totally Disconnected

This chapter evaluates the complements on the pro-étale topology. It addresses two issues raised in the previous lecture on the pro-étale topology. The first issue concerned descent, or more specifically pro-étale descent for perfectoid spaces. The other issue was that the property of being a pro-étale morphism is not local for the pro-étale topology on the target. The chapter then looks at quasi-pro-étale morphisms, as well as G-torsors. A morphism of perfectoid spaces is quasi-pro-étale if for any strictly totally disconnected perfectoid space with a map, the pullback is pro-étale. Using this definition, one can give an equivalent characterization of diamonds.

CHEBYSHEV SETS

2014 ◽  
Vol 98 (2) ◽  
pp. 161-231 ◽  
Author(s):  
JAMES FLETCHER ◽  
WARREN B. MOORS
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Normed Linear Space ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Inner Product Space ◽  
Inner Product ◽  
Best Approximations ◽  
Chebyshev Set ◽  
Chebyshev Sets

AbstractA Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

scholarly journals On the Bishop-Phelps-Bollobás Property for Numerical Radius

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee ◽  
Miguel Martín
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Numerical Radius ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nikodym Property

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show thatL1μ-spaces have this property for every measureμ. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

scholarly journals The fixed point property and normal structure for some B-convex Banach spaces

2001 ◽  
Vol 63 (1) ◽  
pp. 75-81 ◽  
Author(s):  
Jesús García-Falset ◽  
Enrique Llorens-Fuster ◽  
Eva M. Mazcuñán-Navarro
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

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