Zero of 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 $S$.
Then:
- the zero of $\struct{\map {C^*} {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to \R$ defined by:
- $\forall s \in S : \map {0_{\R^S}} s = 0$
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 Zero of Ring of Continuous Real-Valued Functions:
- the zero of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $0_{\R^S} : S \to R$ defined by:
- $\forall s \in S : \map {0_{\R^S}} s = 0$
From Zero of Subring is Zero of Ring:
- $0_{\R^S}$ is the zero of $\struct{\map {C^*} {S, \R}, +, *}$
$\blacksquare$
Also see
Sources
1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$