论文标题
Banach空间的扩展操作员和非线性结构
Extension operators and nonlinear structure of Banach spaces
论文作者
论文摘要
涉及从某个类别扩展功能并根据环境空间的子域定义到整个空间的问题是一个古老而经过良好研究的主题。一个相关的问题是否可以在适当的函数空间之间以线性或连续的方式选择导致过程的扩展，这是高度不平凡的。 That this holds for the class of continuous functions defined on metric spaces is the well-known Borsuk-Dugundji theorem which asserts that given a metric space M and a subspace S of M, each continuous function g on S can be extended to a continuous function f on X such that the resulting assignment from C(S) to C(M) is a norm-one continuous linear extension operator. 本文在Lipschitz函数的可扩展性中，从给定的Banach空间的封闭子空间到整个空间的封闭子空间进行了研究，以使得扩展功能的选择会导致有界的线性（扩展）操作员在Lipschitz功能的适当空间之间。结果表明，当基础空间与希尔伯特空间同构时，指示的属性正是确切的。在该定理的某些有用后果中，我们通过证明闭合Banach空间x的凸出子集时，为X缩回X的Lipsacts缩回X的X缩回时，当X缩回X时X是同异构性的Hilbert空间时，我们提供了一个众所周知的S. Reich定理的同构类似物。我们还将讨论现在在Banach空间的任意子集上定义的Lipschitz函数空间之间的有界线性扩展运算符的问题，并通过使用比作者最初使用的方法更容易访问的方法直接证明了这种扩展运算符的已知不存在。
The problem involving the extension of functions from a certain class and defined on subdomains of the ambient space to the whole space is an old and a well investigated theme in analysis. A related question whether the extensions that result in the process may be chosen in a linear or a continuous manner between appropriate spaces of functions turns out to be highly nontrivial. That this holds for the class of continuous functions defined on metric spaces is the well-known Borsuk-Dugundji theorem which asserts that given a metric space M and a subspace S of M, each continuous function g on S can be extended to a continuous function f on X such that the resulting assignment from C(S) to C(M) is a norm-one continuous linear extension operator. The present paper is devoted to an investigation of this problem in the context of extendability of Lipschitz functions from closed subspaces of a given Banach space to the whole space such that the choice of the extended function gives rise to a bounded linear (extension) operator between appropriate spaces of Lipschitz functions. It is shown that the indicated property holds precisely when the underlying space is isomorphic to a Hilbert space. Among certain useful consequences of this theorem, we provide an isomorphic analogue of a well-known theorem of S. Reich by show ing that closed convex subsets of a Banach space X arise as Lipschitz retracts of X precisely when X is isomorphically a Hilbert space. We shall also discuss the issue of bounded linear extension operators between spaces of Lipschitz functions now defined on arbitrary subsets of Banach spaces and provide a direct proof of the known non-existence of such an extension operator by using methods which are more accessible than those initially employed by the authors.
