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 $\struct {\map C {S, \R}, +, *}$
- $(1) \quad \map {C^*} {S, R} \ne \O$
- $(2) \quad \forall f, g \in \map {C^*} {S, R} : f + \paren{-g} \in \map {C^*} {S, R}$
- $(3) \quad \forall f, g \in \map {C^*} {S, R} : f * g \in \map {C^*} {S, R}$
$(1) \quad \map {C^*} {S, \R} \ne \O$
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, $\S 1.4$