Normed spaces equivalent to inner product spaces and stability of functional equations

Chmieliński, Jacek

doi:10.1007/s00010-013-0193-y

Cited by 7 publications

(2 citation statements)

References 23 publications

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“…In article [24], the author shows that if • i is an equivalent norm coming from an inner product, then the original norm • satisfies an approximate parallelogram law. Following this idea, we can also establish the same results, replace approximate parallelogram law with approximate Euler-Lagrange type identity law.…”

Section: Propositionmentioning

confidence: 99%

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

Liu¹,

Li²

2021

Mathematics

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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.

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Section: Propositionmentioning

confidence: 99%

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

Liu¹,

Li²

2021

Mathematics

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“…Condition (2) in Proposition 4.7 may seem a little restrictive, however, there are many known conditions for a norm being equivalent to a norm generated by an inner product. See for instance [17, Theorem 7.12.68] and [8,Theorem 3.3].…”

Section: Proof Letmentioning

confidence: 99%

Separation axioms and covering dimension of asymmetric normed spaces

Donjuán

Jonard-Pérez

2019

Quaestiones Mathematicae

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It is well known that every asymmetric normed space is a T 0 paratopological group. Since all T i axioms (i = 0, 1, 2, 3) are pairwise nonequivalent in the class of paratopological groups, it is natural to ask if some of these axioms are equivalent in the class of asymmetric normed spaces. In this paper, we will consider this question. We will also show some topological properties of asymmetric normed spaces that are closely related with the axioms T 1 and T 2 (among others). In particular, we will make a remark on [16, Theorem 13], which states that every T 1 asymmetric normed space with compact closed unit ball must be finite-dimensional (as a vector space). We will show that when the asymmetric normed space is finite-dimensional, the topological structure and the covering dimension of the space can be described in terms of certain algebraic properties. In particular, we will characterize the covering dimension of every finite-dimensional asymmetric normed space.2010 Mathematics Subject Classification. 22A30, 46A19, 52A21, 54D10 , 54F45, 54H11,

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Semi-Inner Products and Parapreseminorms on Groups and a Generalization of a Theorem of Maksa and Volkmann on Additive Functions

Száz

2019

Ulam Type Stability

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Normed spaces equivalent to inner product spaces and stability of functional equations

Cited by 7 publications

References 23 publications

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

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

Separation axioms and covering dimension of asymmetric normed spaces

Semi-Inner Products and Parapreseminorms on Groups and a Generalization of a Theorem of Maksa and Volkmann on Additive Functions

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