Question

Let F be an additive group of all continuous functions mapping IR into IR. Let IR be the additive group of real numbers, and let ф :F ------> IR be given by ф(f) = ∫_0^4▒〖f(x)dx〗 . Prove that f is a homomorphism?

Answer 1

Question 1.

Let $F$ be an additive group of all continuous functions mapping $\mathbb{R}$ into $\mathbb{R}$. Let $\mathbb{R}$ be the additive group of real numbers, and let $\phi:F\to\mathbb{R}$ be given by $\phi(f)=\int_{0}^{4}f(x)dx$. Prove that $f$ is a homomorphism.

Solution. Recall that the addition of any two $f,g\in F$ is defined coordinate-wise:

$(f+g)(x)=f(x)+g(x).$

Now taking arbitrary $f,g\in F$ by the additivity of integration we note that

$\phi(f+g)$ $=\int_{0}^{4}(f+g)(x)dx$

$=\int_{0}^{4}(f(x)+g(x))dx$

$=\int_{0}^{4}f(x)dx+\int_{0}^{4}g(x)dx$

$=\phi(f)+\phi(g).$

Thus, $\phi$ is a homomorphism from $F$ to $\mathbb{R}$. ∎