Category:Rings of Bounded Continuous Real-Valued Functions
This category contains results about Rings of Bounded Continuous Real-Valued Functions.
Definitions specific to this category can be found in Definitions/Rings of Bounded Continuous Real-Valued Functions.
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:
- $\forall s \in S : \map {\paren{f g}} s = \map f x \map g s$
Pages in category "Rings of Bounded Continuous Real-Valued Functions"
The following 7 pages are in this category, out of 7 total.
R
- Ring of Bounded Continuous Functions is Ring of Continuous Functions for Compact Space
- Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space
- Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions
- Ring of Bounded Continuous Real-Valued Functions is Commutative