TVS-锥度量空间是度量空间的一个特例
TVS-cone metric spaces as a special case of metric spaces
摘要：
已有许多不动点结果推广到所谓的TVS-锥度量空间，这些空间基于在拓扑向量空间中具有非空内部的锥（体锥）中取值的距离函数。本文证明了TVS-锥度量空间可以配备一族相互等价的（通常的）度量，使得TVS-锥的收敛性（分别为收敛性和收敛性）是一致的。是Cauchy序列的性质，压缩性条件）在TVS意义下等价于收敛性（分别为）。性质是柯西序列，收缩条件）在所有这些度量。因此，我们证明了，如果给定拓扑向量空间$E$和体锥$P$，则TVS-锥度量空间范畴是具有一族相互等价度量的度量空间的真子范畴（推论3.9）。因此，将度量空间的结果推广到TVS-锥度量空间是没有意义的。这也导致了从度量空间到TVS-锥度量空间的不动点结果的形式化推导，并使一些早期的结果变得模糊。在（通常的）度量空间的背景下，我们给出了一个新的公共不动点结果，并证明了它可以很容易地重新表述到TVS-锥度量空间的背景下，尽管形式（句法）推广是不可能的.除了主要结果外，我们还证明了体锥的存在性保证了初始拓扑是Hausdorff的，并且它允许大量的凸开集。事实上，这样的拓扑结构比一些规范拓扑结构更强。
英文摘要：
There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.
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