Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Direct Sum
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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
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Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.18$