Additive Inverse in Ring of Bounded Continuous Real-Valued Functions

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Theorem

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

Let $\R$ denote the real number line.

Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.

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 $\struct{\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.

From Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions:

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

From Additive Inverse in Ring of Continuous Real-Valued Functions:

the additive inverse of $f$ in $\struct{\map C {S, \R}, +, *}$ is the pointwise negation $-f$ defined by:

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

From Inverses in Subgroup:

$-f$ is the additive inverse of $f$

$\blacksquare$

Also see

• Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions

• Zero of Ring of Bounded Continuous Real-Valued Functions

• Unity of Ring of Bounded Continuous Real-Valued Functions

• Ring of Bounded Continuous Real-Valued Functions is Commutative

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$