Definition:Bounded 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 $M' = \struct {Y, \norm {\, \cdot \,}_Y}$ be a normed vector subspace of $M$.


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