bounded

Let $X$ be a subset of $\\mathbb{R}$ . We say that $X$ is bounded when there exists a real number $M$ such that $|x|<M$ for all $x\\in X$ . When $X$ is an interval, we speak of a bounded interval.

This can be generalized first to $\\mathbb{R}^{n}$ . We say that $X\\subseteq\\mathbb{R}^{n}$ is bounded if there is a real number $M$ such that $\\|x\\|<M$ for all $x\\in X$ and $\\|\\cdot\\|$ is the Euclidean distance between $x$ and $y$ .

This condition is equivalent to the statement: There is a real number $T$ such that $\\|x-y\\|<T$ for all $x,y\\in X$ .

A further generalization to any metric space $V$ says that $X\\subseteq V$ is bounded when there is a real number $M$ such that $d(x,y)<M$ for all $x,y\\in X$ , where $d$ is the metric on $V$ .

单词	Bounded1
释义	bounded Let $X$ be a subset of $\\mathbb{R}$ . We say that $X$ is bounded when there exists a real number $M$ such that $\|x\|<M$ for all $x\\in X$ . When $X$ is an interval, we speak of a bounded interval. This can be generalized first to $\\mathbb{R}^{n}$ . We say that $X\\subseteq\\mathbb{R}^{n}$ is bounded if there is a real number $M$ such that $\\\|x\\\|<M$ for all $x\\in X$ and $\\\|\\cdot\\\|$ is the Euclidean distance between $x$ and $y$ . This condition is equivalent to the statement: There is a real number $T$ such that $\\\|x-y\\\|<T$ for all $x,y\\in X$ . A further generalization to any metric space $V$ says that $X\\subseteq V$ is bounded when there is a real number $M$ such that $d(x,y)<M$ for all $x,y\\in X$ , where $d$ is the metric on $V$ .
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