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

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.

Also see

 * Product Rule for Continuous Mappings to Topological Division Ring