Definition:Bounded Normed Vector Space/Definition 1

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

Also see

 * Equivalence of Definitions of Normed Vector Space