Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule

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Theorem

Let $\struct{S, \tau_{_S}}$ be a topological space.

Let $\struct{R, +, *, \norm{\,\cdot\,}}$ be a normed division ring.

Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$.


Let $f, g: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.


Let $f * g : S \to R$ be the mapping defined by:

$\forall x \in S: \map {\paren{f * g}} x = \map f x * \map g x$


Then:

$f * g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is continuous.

Proof

From Corollary to Normed Division Ring Operations are Continuous, $\struct{R, +, *, \tau_{_R}}$ is a topological division ring.

From Product Rule for Continuous Mappings to Topological Division Ring, $f * g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.

$\blacksquare$

Also see