Definition:Bounded Subset of Normed Vector Space
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This page is about Bounded in the context of Normed Vector Space. For other uses, see Bounded.
Definition
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $M' \subseteq X$.
Definition 1
$M'$ is bounded (in $M$) if and only if:
- $\exists x \in X, C \in \R_{> 0}: \forall y \in Y: \norm {x - y} \le C$
Definition 2
$M'$ is bounded (in $M$) if and only if:
- $\exists \epsilon \in \R_{>0} : \exists x \in X : Y \subseteq \map {B_\epsilon^-} x$
where $\map {B_\epsilon^-} x$ is a closed ball in $M$.
Also see
- Equivalence of Definitions of Bounded Subset of Normed Vector Space
- Characterization of von Neumann-Boundedness in Normed Vector Space shows that this notion is equivalent to von Neumann-boundedness