Combination Theorem for Continuous Mappings/Normed Division Ring/Sum 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 Sum Rule for Continuous Mappings to Topological Division Ring:
- $f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
$\blacksquare$