Definition:Open Ball/Normed Vector Space

Definition

Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $x \in X$.

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

The open $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$ is defined as:

$\map {B_\epsilon} x = \set {y \in X: \norm{x - y} < \epsilon}$

Also see

• Definition:Closed Ball in Normed Vector Space

• Definition:Sphere in Normed Vector Space

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.1$: Norms