Definition:Bounded Normed Vector Space/Definition 1

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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:

$\exists x \in X, C \in \R_{> 0}: \forall y \in Y: \norm {x - y} \le C$


Also see