Combination Theorem for Continuous Mappings/Normed Division Ring/Sum 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 Sum Rule for Continuous Mappings to Topological Division Ring:

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

$\blacksquare$

Also see

• Sum Rule for Continuous Mappings to Topological Division Ring