Definition:Ring of Bounded Continuous Real-Valued Functions
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Definition
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 $S$.
The ring of bounded continuous real-valued functions from $S$, denoted $\map {C^*} {S, \R}$, is the set of all bounded continuous mappings in $\map C {S, \R}$ with (pointwise) ring operations $+$ and $*$ restricted to $\map {C^*} {S, R}$.
The (pointwise) ring operations on the ring of bounded continuous real-valued functions from $S$ 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
Sources
1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$