Definition:Ring of Continuous Mappings

Definition

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 $R$.

The ring of continuous mappings from $S$ to $R$, denoted $\map C {S, R}$, is the set of all continuous mappings in $R^S$ with (pointwise) ring operations $+$ and $*$ restricted to $\map C {S, R}$.

The (pointwise) ring operations on the ring of continuous mappings from $S$ to $R$ are defined as:

$\forall f, g \in \map C {S, R} : f + g : S \to R$ is defined by:

$\forall s \in S : \map {\paren{f + g}} s = \map f x + \map g s$

$\forall f, g \in \map C {S, R} : f * g : S \to R$ is defined by:

$\forall s \in S : \map {\paren{f * g}} s = \map f x * \map g s$

Also see

• Ring of Continuous Mappings is Subring of All Mappings

• 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

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