Canonical Injection into Metric Space Product with P-Product Metric is Continuous/Proof 1

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

Let $\pr_1: \MM \to M_1$ and $\pr_2: \MM \to T_2$ be the first and second projections from $\MM$ onto its factors.

From Projection from Metric Space Product with P-Product Metric is Continuous, both $\pr_1$ and $\pr_2$ are continuous

The result follows from Continuous Mapping to Product Space.

$\blacksquare$