V2EX metric space

Metric Space

定义 Definition

Metric space(度量空间):在一个集合上定义了“距离函数”(度量,metric)的数学结构,使得任意两点之间的距离可被量化,并满足非负性同一性对称性三角不等式等基本性质。它是分析、拓扑、几何与机器学习中常见的基础概念。(在更广义的语境中也会讨论“伪度量/半度量”等变体。)

发音 Pronunciation (IPA)

/mtrk spes/

例句 Examples

A circle is a metric space if we measure distances along the arc.
如果我们用沿圆弧的长度来测量距离,圆就可以看作一个度量空间。

Many results in analysis extend naturally from Euclidean space to any complete metric space.
分析学中的许多结论可以从欧几里得空间自然推广到任意完备的度量空间。

词源 Etymology

metric 来自希腊语 metron,意为“测量、尺度”;space 来自拉丁语 spatium,意为“空间、间隔”。合起来的 metric space 字面含义就是“带有测量(距离)规则的空间”。

相关词 Related Words

文学与典籍 Literary Works

  • Topology James R. Munkres(以度量空间作为引入拓扑概念的重要起点)
  • Principles of Mathematical Analysis Walter Rudin(分析学中频繁使用度量空间与完备性等概念)
  • Real and Complex Analysis Walter Rudin(在更高阶分析语境中系统出现)
  • Metric Spaces E. T. Copson(以度量空间为主题的经典教材)
  • A First Course in Functional Analysis John B. Conway(连接度量空间与赋范/巴拿赫空间等内容)
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