Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple 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 $\lambda \in R$.

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


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

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

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

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


Then:

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


Proof

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

From Multiple Rule for Continuous Mappings to Topological Division Ring, $\lambda * f, f * \lambda : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ are continuous mappings.

$\blacksquare$

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