Additive Inverse in Ring of Continuous Mappings

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Theorem

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

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with zero $0_R$.

Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.

Let $f \in \map C {S, R}$.

Then:

the additive inverse of $f$ is the pointwise negation $-f$ defined by:

$\forall s \in S : \map {\paren{-f}} s = - \map f s$

Proof

Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.

From Ring of Continuous Mappings is Subring of All Mappings:

$\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$

From Structure Induced by Ring Operations is Ring:

$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:

$\forall s \in S: \map {\paren {-f} } s := - \map f s$

The result follows.

$\blacksquare$

Also see

• Ring of Continuous Mappings is Subring of All Mappings

• Zero of Ring of Continuous Mappings

• Unity of Ring of Continuous Mappings

• Commutativity of Ring of Continuous Mappings