Direct Product Norm is Norm
Jump to navigation
Jump to search
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 | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {\tuple {x_1, y_1} } + \norm {\tuple {x_2, y_2} }\) |
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces