【作者】 张娜；

【导师】 计东海；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2011， 硕士

【摘要】 在内积空间中,保持正交性的线性算子必是一个线性等距的常数倍。一个很自然的问题是,这个结论在一般的赋范线性空间中是否成立。对保持某种广义正交性的线性算子会使我们对此种广义正交性的性质乃至它对空间性质的影响有一个更深入的研究。首先,本文回顾了各种广义正交性的概念、有关广义正交性的性质、不同种广义正交性之间的关系,保持广义正交的线性算子的已有结论与广义正交有关的内积空间的特征性质。其次,本文介绍了赋范线性空间上的双线性形式、反范数、半内积和上、下半内积的定义,给出了依范数正交和Birkhoff正交、Lumer正交与Birkhoff正交之间的关系。作为本文的主要结论,我们利用反范数给出两个Minkowski平面间线性等距的一个特征性质,证明了一个线性算子T是线性等距的当且仅当Tx的反范数等于x的反范数;一个保持Lumer正交的线性算子必是某个线性等距的常数倍。更多还原

【Abstract】 In inner product spaces, maps preserving orthogonality must be a scalar multiple of a linear isometry. One natural question is whether or not this result still holds in general normed linear spaces. Research on maps preserving some orthogonality helps us to understand properties of this orthogonality and its influence on the properties of the underlying space.Firstly, we survey the introductions of some types of orthogonality, existing results on properties of orthogonality types, relation between two different orthogonality types, and maps preserving generalized orthogonalities, and characterizations of inner product spaces related to generalized orthogonalities.Secondly, we collect the definitions of bilinear form, antinorm, semi-inner product, and the lower and upper semi-inner products, present the relation between norm orthogonality and Birkhoff orthogonality, and between Lumer orthogonality and Birkhoff orthogonality.Finally, as the main results in this paper, we present a characteristic property of linear isometries between two real normed planed in terms of antinorms, prove that a linear map T is a linear isometry if and only if the antinorm of Tx and x are equal , and that a linear map preseving Lumer orthogonality is a scalar multiple of the linear isometry.更多还原