Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 10/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:

$\forall f \in \map {\mathscr C} J: \map \phi f = -\map f 0 + \ds \int_{-2}^0 \map f {e^x} \rd x$


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 $\R$
$m = 1 - e^{-2}$
$I_J$ denotes the identity mapping on $J$.


Proof

From Group Homomorphism: Example 10, 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:

\(\ds \map \phi {f_c}\) \(=\) \(\ds -\map {f_c} 0 + \int_{-2}^0 \map {f_c} {e^x} \rd x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds -c + \int_{-2}^0 c \rd x\) Definition of Constant Mapping
\(\ds \) \(=\) \(\ds -c + \bigintlimits {c x} {-2} 0\) Primitive of Constant
\(\ds \) \(=\) \(\ds -c + \paren {0 - \paren {-2 c} }\)
\(\ds \) \(=\) \(\ds c\)

$\Box$


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

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

where in this case:

$m = 1 - e^{-2}$


We have:

\(\ds \map \phi {I_J - f_m}\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds -\map {\paren {I_J - f_m} } 0 + \int_{-2}^0 \map {\paren {I_J - f_m} } {e^x} \rd x\) \(=\) \(\ds 0\) Definition of $\phi$
\(\ds \leadsto \ \ \) \(\ds -\map {I_J} 0 + \int_{-2}^0 \map {I_J} {e^x} \rd x - \paren {-\map {f_m} 0 + \int_{-2}^0 \map {f_m} {e^x} \rd x}\) \(=\) \(\ds 0\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \leadsto \ \ \) \(\ds -0 + \int_{-2}^0 e^x \rd x - m\) \(=\) \(\ds 0\) Definition of Identity Mapping, a priori
\(\ds \leadsto \ \ \) \(\ds \bigintlimits {e^x} {-2} 0 \rd x\) \(=\) \(\ds m\) Primitive of Exponential Function
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds 1 - e^{-2}\)

$\blacksquare$


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 $13.18 \ \text {(j)}$
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