\begin{definition}[Definition:Induced Norm]
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $Y \subseteq X$ be a subspace.
Then the '''induced norm (on $Y$)''' is defined as the restriction of $\norm {\, \cdot \,}_X$ to $Y$.
:$\norm {\, \cdot \,}_Y := \norm {\, \cdot \,}_X {\restriction_Y}$
\end{definition}