Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 2

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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:

$\forall f \in \map {\mathscr C} J: \map \phi f = \size {\map f 0}$

where $\size {\, \cdot \,}$ denotes the absolute value function.


Then $\phi$ is not a homomorphism.


Proof

Let $f, g \in \map {\mathscr C} J$ be such that:

\(\ds \forall x \in J: \, \) \(\ds \map f x\) \(=\) \(\ds 1\)
\(\ds \forall x \in J: \, \) \(\ds \map g x\) \(=\) \(\ds -1\)

By Constant Real Function is Continuous, both $f$ and $g$ are continuous on $J$.


We have:

\(\ds \map \phi {f + g}\) \(=\) \(\ds \size {\map {\paren {f + g} } 0}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \size {\map f 0 + \map g 0}\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \) \(=\) \(\ds \size {1 + \paren {-1} }\) Definition of $f$ and $g$
\(\ds \) \(=\) \(\ds \size 0\) arithmetic
\(\ds \) \(=\) \(\ds 0\) Definition of Absolute Value

However:

\(\ds \map \phi f + \map \phi g\) \(=\) \(\ds \size {\map f 0} + \size {\map g 0}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \size 1 + \size {-1}\) Definition of $f$ and $g$
\(\ds \) \(=\) \(\ds 1 + 1\) Definition of Absolute Value
\(\ds \) \(=\) \(\ds 2\) arithmetic

Thus:

$\map \phi {f + g} \ne \map \phi f + \map \phi g$

and so $\phi$ 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 $13.18 \ \text {(b)}$
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