Definition:Closed Ball/Normed Vector Space
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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 closed $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$ is defined as:
- $\map { {B_\epsilon}^-} x = \set {y \in X: \norm {y - x} \le \epsilon}$
Also see
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.1$: Norms