Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space

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

Let $Y \subseteq X$ be a vector subspace.

Let $\norm {\, \cdot \,}_Y$ be the induced norm on $Y$.

Then $\struct {Y, \norm {\, \cdot \,}_Y}$ is a normed vector space.

Positive definiteness
By definition of induced norm:


 * $\forall y \in Y : \norm {y}_Y = \norm {y}_X > 0$

Suppose $y \in Y : \norm {y}_Y = 0$.

Since $\norm {\, \cdot \,}_Y$ is an induced norm in $\struct {X, \norm {\, \cdot \,}_X}$:


 * $\norm {y}_X = 0$.

Therefore, $y = \mathbf 0 \in X$.

By definition of a vector subspace, $Y$ has the additive identity:


 * $\mathbf 0 \in Y$.

Thus:


 * $y = \mathbf 0 \in Y \subseteq X$.

Positive homogeneity
Let $y \in Y$.

Let $\alpha \in \R$.

Then we have that:

Triangle inequality
Let $y_1, y_2 \in Y$.

By closure axiom of vector space:


 * $y_1 + y_2 \in Y$.

Furthermore: