# 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| 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\| 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\| 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) for all $x,y\in X$, where $d$ is the metric on $V$.

Title bounded Bounded1 2013-03-22 14:00:00 2013-03-22 14:00:00 yark (2760) yark (2760) 11 yark (2760) Definition msc 54E35 EuclideanDistance MetricSpace bounded interval