Banach空间等距及其扰动理论的研究

The research on theory of isometries and their perturbations of Banach space has a long history, rich achievements and far-reaching influence. This project is committed to the following research. (1) We will explore linear properties of perturbed isometric embedding of Banach space, so as to describe the characterizations such that the existence of a perturbed isometric embedding implies the existence of an isometric embedding, even the existence of a linear isometric embedding; (2) Based on the structure and properties of the unit ball of Banach space, we will investigate the representation subset theory of isometric embedding, and reveal the essential relationship between the existence of a locally isometric embedding, the existence of an isometric embedding of the whole space and the existence of a linear isometric embedding of the whole space; (3) We will study the characterizations such that perturbed isometric embedding can be uniformly approximated by isometric embedding, and estimate the optimal error. (4) Making use of the duality theory, the invariant averaging technique and the projection operators of Banach space, we will research the necessary and sufficient conditions for the existence of the linear left-quasi inverse of the perturbed isometric embedding. (5) We will discuss the representation theory of perturbed isometric embedding on normed semi-groups, which are generated by convex subsets of Banach space, and clarify the intrinsic relationship between linear structure of Banach spaces and the metric structure of normed semi-groups. The realization of the above goals can reveal the essential connection and difference between the linear structure and the metric structure of Banach space, and provide reference and guidance for the study of nonlinear geometric theory of Banach space.

Banach空间等距及其扰动理论的研究具有悠久历史、丰硕成果和深远影响。本项目致力于（1）挖掘Banach空间扰动等距嵌入所蕴含的线性性质，刻画空间扰动等距嵌入蕴含空间等距嵌入、甚至线性等距嵌入的特征；（2）利用Banach空间单位球的结构和性质，研究等距嵌入的表示子集理论，揭示空间局部等距嵌入、全空间的等距嵌入和全空间的线性等距嵌入之间的本质关系；(3）研究Banach空间中扰动等距嵌入可以被等距嵌入一致逼近的特征，并估计一致逼近的最佳误差；（4）利用Banach空间对偶理论、不变平均技巧、投影算子等思想方法，研究扰动等距嵌入的线性左拟逆存在的充要条件；（5）研究Banach空间凸子集生成的赋范半群上扰动等距嵌入的表示，阐明赋范半群的度量结构与空间线性结构之间的内在联系。上述目标的实现可以揭示Banach空间线性结构与度量结构的本质联系和区别，为空间非线性几何理论的研究提供借鉴和指导。

根据本项目申报书和计划书拟定的各项安排，项目组对Banach空间等距及其扰动理论开展了广泛深刻的研究。主要研究内容、重要结果及其科学意义体现如下。首先，我们研究Banach空间扰动等距映射的一致逼近理论，证明了当像空间为连续函数空间时，任意扰动等距映射可被等距映射一致逼近，因此推广了前人著名的结论，并在此条件下肯定地解决了Hyers-Ulam稳定性问题。其次，我们研究Banach空间紧凸子集构成的Hausdorff度量空间之间的等距映射的表示理论，证明了特殊Banach空间（包含所有光滑Banach空间）的紧凸子集构成的Hausdorff度量空间之间的满等距映射可表示成底空间之间的线性满等距的自然延拓，证明了有限维Banach空间紧凸子集构成的Hausdorff度量空间之间的可加满等距映射可表示成底空间之间的线性满等距的自然延拓，因此揭示了紧凸子集Hausdorff度量空间的结构与Banach空间的结构的本质联系。再次，我们研究Banach空间有界闭凸子集构成的Hausdorff度量空间之间的等距映射的表示理论，证明了Banach空间特殊有界闭凸子集构成的Hausdorff度量空间之间的可加满等距映射可表示成底空间之间的线性满等距的自然延拓，因此揭示了特殊有界闭凸子集Hausdorff度量空间的结构与Banach空间的结构的深刻联系。最后，我们研究Banach空间紧凸子集构成的Hausdorff度量空间之间的扰动等距映射的一致逼近理论，证明了特殊Banach空间（包含所有光滑Banach空间）的紧凸子集构成的Hausdorff度量空间之间的双射的扰动等距映射可被等距映射一致逼近，获得逼近误差的最优估计，因此在此条件下肯定地解决了Hyers-Ulam稳定性问题。