Complements on Diminnie Orthogonality

A new orthogonality relation for normed linear spaces is introduced by C. R. DIMINNIE in [10]. Some interesting properties of such orthogonality and its relationship with Birkhoff orthogonality are studied in the above paper. The first part of this paper begins with a geometrical interpretation of Diminnie-orthogonality which allows us to obtain some other properties of such orthogonality. The second part deals with relationships between Diminnie orthogonality and some other known orthogonalities.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

Area orthogonality in normed linear spaces

A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.     

An orthogonality type based on invariant inner products

Aequationes mathematicae - A new orthogonality type in normed linear spaces, which is based on invariant inner products, is introduced. It is shown that this orthogonality has properties of...     

A characterization of inner product spaces related to the skew-angular distance

A new refinement of the triangle inequality is presented in normed linear spaces. Moreover, a simple characterization of inner product spaces is obtained by using the skew-angular distance.     

Orthogonalities,linear operators,and characterization of inner product spaces

It is proved that a normed space, whose dimension is at least three, admitting a nonzero linear operator reversing Birkhoff orthogonality is an inner product space, which releases the smoothness condition in one of J. Chmieliński’s results. Further characterizations of inner product spaces are obtained by studying properties of linear operators related to Birkhoff orthogonality and isosceles orthogonality.     

Spectral Theory for Riemannian Manifolds with Cusps and a Related Trace Formula

A new orthogonality relation for normed linear spaces is introduced using a concept of area of a parallelogram given by E. Silverman. Comparisons are drawn between this relation and an earlier relation used by G. Birkhoff. In addition, this new relation is utilized to obtain new characterizations of inner-product spaces.     

Isosceles-orthogonality preserving property and its stability

For real normed spaces, we consider the class of linear operators, preserving approximately the relation of isosceles-orthogonality. We show some general properties of such mappings. Next, we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one. In regard to this, we generalize some results obtained earlier for inner product spaces with standard orthogonality relation.     

An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

A new orthogonality relation in normed linear spaces which generalizes pythagorean orthogonality and isosceles orthogonality is defined, and it is shown that the new orthogonality is homogeneous (additive) if and only if the space is a real inner-product space.     

Linear approximation of approximately biorthogonality preserving maps

We investigate linear properties of mappings from a bounded domain of an n-dimensional normed space into another n-dimensional normed space such that the image of some almost biorthogonal system is almost biorthogonal. In this way we generalize a result of the author on stability of orthogonality in Euclidean spaces.     

Metrizations of orthogonality and characterizations of inner product spaces

Characterizations of real inner product spaces among normed linear spaces have been obtained by exploring properties of and relationships between various orthogonality relations which can be defined in such spaces. In the present paper the authors present metrized versions of some of these properties and relationships and obtain new characterizations of real inner product spaces among complete, convex, externally convex metric spaces.     

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.     

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·) _i and (·, ·) _S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.     

关于随机赋范空间与随机内积空间的某些基本理论(英文)

首先提出随机度量空间定义的另一个提法,这提法不仅等价于原始的定义而且也使随机度量空间自动归入广义度量空间的框架,也考虑了关于拓扑结构的某些新的问题;循着同样的思路,对随机赋范空间的定义也作了新的处理并同时简化了随机赋范模的定义．其次本文也证明了一个Ｅ－范空间的商空间等距同构于一个典型的Ｅ－范空间;进一步,在概率赋范空间的框架下证明了一个概率赋伪范空间是伪内积生成空间的充要条件是它等距同构于一个Ｅ－内积空间,这回答了Ｃ．Ａｌｓｉｎａ与Ｂ．Ｓｃｈｗｅｉｚｅｒ等人新近提出的公开问题．最后,本文转向了它的中心部分──关于随机内积空间的研究,对随机内积空间中的特有且复杂的正交性作较系统的讨论,论证了只有几乎处处正交性才是唯一合理的正交性概念,在此基础上本文尤其将Ｇ．Ｓｔａｍｐａｃｃｈｉａ的在众多学科中都有多种用途的一般投影定理（或称变分不等式解存在性定理）以适当形式推广到完备实随机内积模上．     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.     

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

Strong additivity of metric isosceles orthogonality

It is known that the property of additivity of isosceles orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper it is shown that suitably metrized concepts of additivity of metric isosceles orthogonality characterize euclidean or hyperbolic spaces among complete, convex, externally convex metric spaces.     

A generalized orthogonality relation via norm derivatives in real normed linear spaces

Aequationes mathematicae - We introduce a generalized orthogonality relation in real normed linear spaces via norm derivatives. The relation between this concept and other types of orthogonalities...