Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 8/Kernel
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:
- $\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \int_0^1 \map f {x y} \rd y \rd x$
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 $\R$
- $m = \dfrac 1 4$
- $I_J$ denotes the identity mapping on $J$.
Proof
From Group Homomorphism: Example 8, 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:
\(\ds \map \phi {f_c}\) | \(=\) | \(\ds \int_0^1 \int_0^1 \map {f_c} {x y} \rd y \rd x\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \int_0^1 c \rd y \rd x\) | Definition of Constant Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \bigintlimits {c y} 0 1 \rd x\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 c \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigintlimits {c x} 0 1\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds c\) |
$\Box$
Then we show that there exists a unique $m \in \R$ such that:
- $\map \phi {I_J - f_m} = 0$
where in this case:
- $m = \dfrac 1 4$
We have:
\(\ds \map \phi {I_J - f_m}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^1 \int_0^1 \map {\paren {I_J - f_m} } {x y} \rd y \rd x\) | \(=\) | \(\ds 0\) | Definition of $\phi$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^1 \int_0^1 \map {I_J} {x y} \rd y \rd x - \int_0^1 \int_0^1 \map {f_m} {x y} \rd y \rd x\) | \(=\) | \(\ds 0\) | Definition of Pointwise Addition of Real-Valued Functions | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^1 \int_0^1 x y \rd y \rd x - m\) | \(=\) | \(\ds 0\) | Definition of Identity Mapping, a priori | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^1 \intlimits {\dfrac {x y^2} 2} {y \mathop = 0} {y \mathop = 1} \rd x\) | \(=\) | \(\ds m\) | Primitive of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^1 \dfrac x 2 \rd x\) | \(=\) | \(\ds m\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \intlimits {\dfrac {x^2} 4} {x \mathop = 0} {x \mathop = 1}\) | \(=\) | \(\ds m\) | Primitive of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds m\) | \(=\) | \(\ds \dfrac 1 4\) |
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 $13.18 \ \text {(h)}$