Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 1/Kernel

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Example of Group Homomorphism

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

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

Let $I_J$ denote the identity mapping on $J$:

$\forall x \in J: \map {I_J} x = x$


Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the homomorphism defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \map f 1$


The kernel of $\phi$ is given by:

$\map \ker \phi = I_J - f_m$

where:

$f_m: \R \to \R$ denotes the constant mapping on $\R$
$m = 1$
$I_J$ denotes the identity mapping on $J$.


Proof

From Group Homomorphism: Example 1, we have that $\phi$ is indeed a homomorphism.

For all $c \in \R$, let $f_c: \R \to \R$ be the constant mapping:

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


First we show that:

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


Let $c \in \R$ be arbitrary.

We have:

\(\ds \map \phi {f_c}\) \(=\) \(\ds \map {f_c} 1\) Definition of $\phi$
\(\ds \) \(=\) \(\ds c\) Definition of Constant Mapping

$\Box$


Then we show that there exists a unique $m \in \R$ such that:

$\map \phi {I_J - f_m} = 0$

where in this case:

$m = 1$


We have:

\(\ds \map \phi {I_J - f_m}\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \map {\paren {I_J - f_m} } 1\) \(=\) \(\ds 0\) Definition of $\phi$
\(\ds \leadsto \ \ \) \(\ds \map {I_J} 1 + \paren {-\map {f_m} 1}\) \(=\) \(\ds 0\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \leadsto \ \ \) \(\ds 1 + \paren {-m}\) \(=\) \(\ds 0\) Definition of Identity Mapping, Definition of Constant Mapping
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds 1\)

Hence the result by definition of kernel.

$\blacksquare$


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 $13.18 \ \text {(a)}$
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