Definition:Closed Ball

From ProofWiki
Jump to: navigation, search

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

Let $\epsilon \in \R_{>0}$ be a positive real number.


The closed $\epsilon$-ball of $a$ in $M$ is defined as:

${B_\epsilon}^- \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) \le \epsilon}\right\}$

where $B^-$ recalls the notation of topological closure.


If it is necessary to show the metric itself, then the notation ${B_\epsilon}^- \left({a; d}\right)$ can be used.


Radius

In ${B_\epsilon}^- \left({a}\right)$, the value $\epsilon$ is referred to as the radius of the closed $\epsilon$-ball.


Normed Division Ring

Let $\struct{R, \norm{\,\cdot\,}}$ be a normed division ring.

Let $a \in R$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.


The closed $\epsilon$-ball of $a$ in $\struct{R, \norm{\,\cdot\,}}$ is defined as:

${B_\epsilon}^- \paren{a} = \set {x \in R: \norm{x - a} \le \epsilon}$


Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

By the definition of the metric induced by the norm, the closed $\epsilon$-ball of $a$ in $\struct{R, \norm{\,\cdot\,}}$ is the closed $\epsilon$-ball of $a$ in $\struct{R, d}$


If it is necessary to show the norm itself, then the notation ${B_\epsilon}^- \paren{a; \norm{\,\cdot\,}}$ can be used.


Real Analysis

Let $n \ge 1$ be a natural number.

Let $\R^n$ denote real Euclidean space

Let $\left\Vert{\, \cdot \,}\right\Vert$ denote the Euclidean norm.

Let $a \in \R^n$.

Let $R > 0$ be a strictly positive real number.


The closed ball of center $a$ and radius $R$ is the subset:

$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert \le R}\right\}$


Also denoted as

The notation $B^- \left({a; \epsilon}\right)$ can be found for ${B_\epsilon}^- \left({a}\right)$, particularly when $\epsilon$ is a more complicated expression than a constant.

Similarly, some sources allow ${B_d}^- \left({a; \epsilon}\right)$ to be used for ${B_\epsilon}^- \left({a; d}\right)$.

It needs to be noticed that the two styles of notation allow a potential source of confusion, so it is important to be certain which one is meant.


Also see


Sources