Direct Product Norm is Norm

Theorem
Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces.

Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations.

Let $\norm {\tuple {x, y} }$ be the direct product norm.

Then $\norm {\tuple {x, y} }$ is a norm on $V$.

Positive Definiteness
Let $\tuple {x, y} \in V$.

Then:

Suppose $\norm {\tuple {x, y} } = 0$.

Then:

Hence, $\norm x = 0$

Triangle Inequality
Together:


 * $\map \max {\norm {x_1 + x_2}, \norm {y_1 + y_2}} \le \norm {\tuple {x_1, y_1} } + \norm {\tuple {x_2, y_2} }$

Then: