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

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

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


From Pointwise Addition on Continuous Real Functions on Closed Unit Interval forms Group we have that $G$ is indeed a group.


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

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


Then $\phi$ is a homomorphism.


Kernel

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

Let $f, g \in \map {\mathscr C} J$ be arbitrary.

We note that:

when $x = 0$ we have that $e^x = 1$
when $x = -2$ we have that $0 < e^x < 1$

Therefore the limits of integration of the given definition of $\phi$ are within $\closedint 0 1$.

As both $f$ and $g$ are continuous real functions on $J$, they are integrable on $\closedint {e^{-2} } 1$.

We have:

\(\ds \map \phi {f + g}\) \(=\) \(\ds -\map {\paren {f + g} } 0 + \ds \int_{-2}^0 \map {\paren {f + g} } {e^x} \rd x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds -\map f 0 - \map g 0 + \ds \int_{-2}^0 \paren {\map f {e^x} + \map g {e^x} } \rd x\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \) \(=\) \(\ds -\map f 0 + \ds \int_{-2}^0 \map f {e^x} \rd x - \map g 0 + \ds \int_{-2}^0 \map g {e^x} \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \map \phi f + \map \phi g\) Definition of $\phi$

Thus $\phi$ is a homomorphism by definition.

$\blacksquare$


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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