Canonical Injection into Metric Space Product with P-Product Metric is Continuous/Proof 2
Theorem
Let $M_1 = \struct {A_1, d}$ and $M_2 = \struct {A_2, d'}$ be metric spaces.
Let $\AA := A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d_p}$ denote the metric space on $\AA$ where $d_p: \AA \to \R$ is one of the $p$-product metrics on $\AA$:
- $\map {d_p} {x, y} := \begin {cases} \paren {\paren {\map d {x_1, y_1} }^p + \paren {\map {d'} {x_2, y_2} }^p}^{1/p} & : p \in \Z_{>0} \\ \ds \max_{i \mathop = 1}^n \set {\map d {x_1, y_1}, \map {d'} {x_2, y_2} } & : p = \infty \end {cases}$
where:
- $x = \tuple {x_1, x_2}$
- $y = \tuple {y_1, y_2}$
Let $a \in A_1$ and $b \in A_2$ be fixed and arbitrary.
Let:
- $i_b: A_1 \to \AA$ be the mapping defined as:
- $\forall x \in A_1: \map {i_b} x = \tuple {x, b}$
- $i_a: A_2 \to \AA$ be the mapping defined as:
- $\forall y \in A_2: \map {i_a} y = \tuple {a, y}$
Then $i_b$ and $i_a$ are continuous in $M_1$ and $M_2$ respectively.
Proof
We want to show that:
- $\forall c_1 \in A_1: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map d {x, c} < \delta \implies \map {d_p} {\map {i_b} x, \map {i_b} c} < \epsilon$
and:
- $\forall c_2 \in A_2: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d'} {x, c} < \delta \implies \map {d_p} {\map {i_a} c, \map {i_a} x} < \epsilon$
Let $c_1$ and $c_2$ in $A_1$ and $A_2$ respectively be arbitrary.
Let $\epsilon \in \R_{>0}$ also be arbitrary.
Let $\delta = \epsilon$.
Let $x \in A_1$ such that $\map d {x, c_1} < \delta$.
We have:
\(\ds \map {d_p} {\map {i_b} x, \map {i_b} {c_1} }\) | \(=\) | \(\ds \map {d_p} {\tuple {x, b}, \tuple {c_1, b} }\) | Definition of $i_b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} \paren {\paren {\map d {x, {c_1} } }^p + \paren {\map {d'} {b, b} }^p}^{1/p} & : p \in \Z_{>0} \\ \ds \max_{i \mathop = 1}^n \set {\map d {x, c_1}, \map {d'} {b, b} } & : p = \infty \end {cases}\) | Definition of $d_p$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} \paren {\paren {\map d {x, {c_1} } }^p + 0^p}^{1/p} & : p \in \Z_{>0} \\ \ds \max_{i \mathop = 1}^n \set {\map d {x, {c_1} }, 0} & : p = \infty \end {cases}\) | Metric Space Axiom $(\text M 1)$ applied to $\map {d'} {b, b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {x, {c_1} }\) | after simplification | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) | as $\epsilon = \delta$ |
We have that $c_1$ and $\epsilon$ are arbitrary.
Hence, by definition, $i_b$ is continuous in $M_1$.
$\Box$
Let $y \in A_2$ such that $\map d {y, c_2} < \delta$.
We have:
\(\ds \map {d_p} {\map {i_a} y, \map {i_a} {c_2} }\) | \(=\) | \(\ds \map {d_p} {\tuple {a, y}, \tuple {a, c_2} }\) | Definition of $i_a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} \paren {\paren {\map d {a, a} }^p + \paren {\map {d'} {y, c_2} }^p}^{1/p} & : p \in \Z_{>0} \\ \ds \max_{i \mathop = 1}^n \set {\map d {a, a}, \map {d'} {y, c_2} } & : p = \infty \end {cases}\) | Definition of $d_p$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin {cases} \paren {0^p + \paren {\map {d'} {y, c_2} }^p}^{1/p} & : p \in \Z_{>0} \\ \ds \max_{i \mathop = 1}^n \set {0, \map {d'} {y, c_2} } & : p = \infty \end {cases}\) | Metric Space Axiom $(\text M 1)$ applied to $\map d {a, a}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map d {y, {c_2} }\) | after simplification | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) | as $\epsilon = \delta$ |
We have that $c_2$ and $\epsilon$ are arbitrary.
Hence, by definition, $i_a$ is continuous in $M_2$.
$\blacksquare$