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

Examples of Group Homomorphisms

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.

Direct Sum

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be a homomorphism from $\struct {\map {\mathscr C} J, +}$ to $\struct {\R, +}$.

Let $\phi$ satisfy the condition:

$\forall c \in \R: \map \phi {f_c} = c$

where $f_c$ is the constant mapping on $\R$ defined as:

$\forall x \in \R: \map {f_c} x = c$

Then $\struct {\map {\mathscr C} J, +}$ is the internal group direct product of $\map \ker \phi$ and the subgroup of constant mappings on $\R$.

Example $1$

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

$\forall f \in \map {\mathscr C} J: \map \phi f = \map f 1$

Then $\phi$ is a homomorphism.

Example $2$

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.

Example $3$

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

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

Then $\phi$ is a homomorphism.

Example $4$

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

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \dfrac \pi 3 \int_0^1 \map f x \cos \dfrac {\pi x} 6 \rd x$

Then $\phi$ is a homomorphism.

Example $5$

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

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map \cos {\dfrac {\pi \map f x} 6} \rd x$

Then $\phi$ is not a homomorphism.

Example $6$

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

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map f {\cos \dfrac {\pi x} 6} \rd x$

Then $\phi$ is a homomorphism.

Example $7$

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.

Example $8$

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

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

Then $\phi$ is a homomorphism.

Example $9$

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

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds 2 \int_0^1 \int_0^x \map f y \rd y \rd x$

Then $\phi$ is a homomorphism.

Example $10$

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.

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 $13.18$