Restriction of Norm on Vector Space to Subspace is Norm

Theorem
Let $\Bbb F$ be a subfield of $\C$.

Let $X$ be a vector space over $\Bbb F$.

Let $\norm \cdot_X : X \to \hointr 0 \infty$ be a norm on $X$.

Let $Y$ be a vector subspace of $Y$.

Then $\norm \cdot_Y$, the restriction of $\norm \cdot_X$ to $Y$ is a norm on $Y$.

Proof
Since $\norm x_X \ge 0$ for any $x \in X$, we have $\norm y_Y \ge 0$ for any $y \in Y$.

We verify each of the axioms for a norm on a vector space.

Proof of $(\text N 1)$
Note that for $y \in Y$ we have $\norm y_Y = 0$ $\norm y_X = 0$.

Then by $(\text N 1)$ for $\norm \cdot_X$, we have $y = 0$.

Proof of $(\text N 2)$
Let $\lambda \in \Bbb F$ and $y \in Y$.

Then, we have:

Proof of $(\text N 3)$
Let $x, y \in Y$.

Then, we have: