Ring of Continuous Mappings is Subring of All Mappings

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Theorem

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

Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring.

Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.

Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.

Then:

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

Proof

From Structure Induced by Ring Operations is Ring:

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

From Structure Induced by Ring Operations is Ring:

$\forall f \in R^S :$ 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 $\struct {R^S, +, *}$

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 $0_R$ denote the zero of $\struct {R, +, *, \tau_{_R} }$.

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

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

From Constant Mapping is Continuous:

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

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 Continuous Mappings to Topological Ring:

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

From Sum Rule for Continuous Mappings into Topological Ring:

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

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 Continuous Mappings to Topological Ring:

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

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 $\struct {R^S, +, *}$.

$\blacksquare$

Also see

• Zero of Ring of Continuous Mappings

• Additive Inverse in Ring of Continuous Mappings

• Unity of Ring of Continuous Mappings

• Commutativity of Ring of Continuous Mappings