Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Direct Sum

From ProofWiki
Jump to navigation Jump to search

Example of Group Homomorphisms

Let $J \subseteq \R$ denote the closed unit interval $\closedint 0 1$.

Let $\map {\mathscr C} J$ denote the set of all continuous real functions from $J$ to $\R$.

Let $G = \struct {\map {\mathscr C} J, +}$ be the group formed on $\map {\mathscr C} J$ by pointwise addition.

Let $\struct {\R, +}$ denote the additive group of real numbers.


Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be a homomorphism from $\struct {\map {\mathscr C} J, +}$ to $\struct {\R, +}$.

Let $\phi$ satisfy the condition:

$\forall c \in \R: \map \phi {f_c} = c$

where $f_c$ is the constant mapping on $\R$ defined as:

$\forall x \in \R: \map {f_c} x = c$


Then $\struct {\map {\mathscr C} J, +}$ is the internal group direct product of $\map \ker \phi$ and the subgroup of constant mappings on $\R$.


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Group_Homomorphism/Examples/Pointwise_Addition_on_Continuous_Real_Functions_on_Closed_Unit_Interval/Direct_Sum&oldid=566706"