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