Direct Product Norm is Norm

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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 $V$.


Proof

Positive Definiteness

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

Then:

\(\ds \norm {\tuple {x, y} }\) \(=\) \(\ds \map \max {\norm x, \norm y}\) Definition of Direct Product Norm
\(\ds \) \(\ge\) \(\ds 0\) Norm Axiom $N1$: Positive Definiteness


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

Then:

\(\ds 0\) \(\le\) \(\ds \norm x\) Norm Axiom $N1$: Positive Definiteness
\(\ds \) \(\le\) \(\ds \map \max {\norm x, \norm y}\)
\(\ds \) \(=\) \(\ds \norm {\tuple {x, y} }\) Definition of Direct Product Norm
\(\ds \) \(=\) \(\ds 0\)

Hence, $\norm x = 0$

$\Box$


Positive Homogeneity

\(\ds \norm {\alpha \tuple {x, y} }\) \(=\) \(\ds \norm {\tuple {\alpha x, \alpha y} }\) Induced component-wise operations
\(\ds \) \(=\) \(\ds \map \max {\norm {\alpha x}, \norm {\alpha y} }\) Definition of Direct Product Norm
\(\ds \) \(=\) \(\ds \map \max {\size \alpha \norm x, \size \alpha \norm y}\) Norm Axiom $N2$ : Positive Homogeneity
\(\ds \) \(=\) \(\ds \size \alpha \max \set {\norm x, \norm y}\)
\(\ds \) \(=\) \(\ds \size \alpha \norm {\tuple {x ,y} }\) Definition of Direct Product Norm

$\Box$


Triangle Inequality

\(\ds \norm {x_1 + x_2}\) \(\le\) \(\ds \norm {x_1} + \norm {x_2}\)
\(\ds \) \(\le\) \(\ds \map \max {\norm {x_1}, \norm {y_1} } + \map \max {\norm {x_2}, \norm {y_2} }\)
\(\ds \) \(=\) \(\ds \norm {\tuple {x_1, y_1} }+ \norm {\tuple {x_2, y_2} }\) Definition of Direct Product Norm
\(\ds \norm {y_1 + y_2}\) \(\le\) \(\ds \norm {y_1} + \norm {y_2}\)
\(\ds \) \(\le\) \(\ds \map \max {\norm {x_1}, \norm {y_1} } + \map \max {\norm {x_2}, \norm {y_2} }\)
\(\ds \) \(=\) \(\ds \norm {\tuple {x_1, y_1} } + \norm {\tuple {x_2, y_2} }\) Definition of Direct Product Norm

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:

\(\ds \norm {\tuple {x_1, y_1} + \tuple {x_2, y_2} }\) \(=\) \(\ds \norm {\tuple {x_1 + x_2, y_1 + y_2} }\) Induced component-wise operations
\(\ds \) \(=\) \(\ds \map \max {\norm {x_1 + x_2}, \norm {y_1 + y_2} }\) Definition of Direct Product Norm
\(\ds \) \(\le\) \(\ds \norm {\tuple {x_1, y_1} } + \norm {\tuple {x_2, y_2} }\)

$\blacksquare$


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