Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 7

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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:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \int_0^1 \map f x \map f y \rd y \rd x$


Then $\phi$ is not a homomorphism.


Proof

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

As both $f$ and $g$ are continuous real functions on $J$, they are integrable on $\closedint 0 1$.

We have:

\(\ds \map \phi {f + g}\) \(=\) \(\ds \int_0^1 \int_0^1 \map {\paren {f + g} } x \map {\paren {f + g} } y \rd y \rd x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \int_0^1 \int_0^1 \paren {\map f x + \map g x} \paren {\map f y + \map g y} \rd y \rd x\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \) \(=\) \(\ds \int_0^1 \int_0^1 \paren {\map f x \map f y + \map g x \map f y + \map f x \map g y + \map g x \map g y} \rd y \rd x\)
\(\ds \) \(=\) \(\ds \int_0^1 \int_0^1 \map f x \map f y \rd y \rd x + \int_0^1 \int_0^1 \map g x \map f y \rd y \rd x + \int_0^1 \int_0^1 \map f x \map g y \rd y \rd x + \int_0^1 \int_0^1 \map g x \map g y \rd y \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(\ne\) \(\ds \int_0^1 \int_0^1 \map f x \map f y \rd y \rd x + \int_0^1 \int_0^1 \map g x \map g y \rd y \rd x\)
\(\ds \) \(=\) \(\ds \map \phi f + \map \phi g\) Definition of $\phi$

Thus $\phi$ is not 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 $13.18 \ \text {(g)}$
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