Unity of Ring of Bounded Continuous Real-Valued Functions

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

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 unity of $\struct{\map {C^*} {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to \R$ defined by:

$\forall s \in S : \map {1_{\R^S}} s = 1$.

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 Unity of Ring of Continuous Real-Valued Functions:

the unity of $\struct{\map C {S, \R}, +, *}$ is the constant mapping $1_{\R^S} : S \to R$ defined by:

$\forall s \in S : \map {1_{\R^S}} s = 1$

From Constant Mapping is Continuous and Constant Real-Valued Function is Bounded:

$1_{\R^S} \in \map {C^*} {S, \R}$

From Subring Containing Ring Unity has Unity:

$1_{\R^S}$ is the unity of $\struct{\map {C^*} {S, \R}, +, *}$

$\blacksquare$

Also see

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

• Zero of Ring of Bounded Continuous Real-Valued Functions

• Additive Inverse in 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, $\S 1.4$