Definition:Bounded Normed Vector Space/Definition 2

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Definition

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

Let $M' = \struct {Y, \norm {\, \cdot \,}_Y}$ be a subspace of $M$.


$M'$ 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