Category:Rings of Continuous Mappings
This category contains results about Rings of Continuous Mappings.
Definitions specific to this category can be found in Definitions/Rings of Continuous Mappings.
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:
- $\forall s \in S : \map {\paren{f * g}} s = \map f x * \map g s$
Subcategories
This category has only the following subcategory.
R
Pages in category "Rings of Continuous Mappings"
The following 5 pages are in this category, out of 5 total.