Ring of Bounded Continuous Functions is Subring of 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 continuous real-valued functions from $S$.

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


Then:

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


Proof

From Ring of Continuous Real-Valued Functions is Ring:

$\struct {\map C {S, \R}, +, *}$ is a ring.


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

$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
$\forall s \in S: \map {\paren {-f} } s := - \map f s$


From the Subring Test:

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

if and only if

$(1) \quad \map {C^*} {S, R} \ne \O$
$(2) \quad \forall f, g \in \map {C^*} {S, R} : f + \paren{-g} \in \map {C^*} {S, R}$
$(3) \quad \forall f, g \in \map {C^*} {S, R} : f * g \in \map {C^*} {S, R}$


$(1) \quad \map {C^*} {S, \R} \ne \O$

Let $0_{\R^S}: S \to \R$ denote the constant mapping defined by:

$\forall s \in S : \map {0_{\R^S}} s = 0$

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

$0_{\R^S} \in \map {C^*} {S, \R}$

It follows that:

$\map {C^*} {S, \R} \ne \O$

$\Box$


$(2) \quad \forall f, g \in \map {C^*} {S, \R} : f + \paren{-g} \in \map {C^*} {S, \R}$

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


From Negation Rule for Bounded Continuous Real-Valued Function:

$-g \in \map {C^*} {S, \R}$

From Sum Rule for Bounded Continuous Real-Valued Functions:

$f + \paren{-g} \in \map {C^*} {S, \R}$


It follows that:

$\forall f, g \in \map {C^*} {S, \R} : f + \paren{-g} \in \map C {S, R}$

$\Box$


$(3) \quad \forall f, g \in \map {C^*} {S, \R} : f * g \in \map {C^*} {S, \R}$

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


From Product Rule for Bounded Continuous Real-Valued Functions:

$f * g \in \map {C^*} {S, \R}$


It follows that:

$\forall f, g \in \map {C^*} {S, \R} : f * g \in \map {C^*} {S, \R}$

$\Box$


From Subring Test:

$\struct{\map {C^*} {S, \R}, +, *}$ is a subring 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, $\S 1.4$

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