Subspace of Riemann Integrable Functions

Theorem
Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$.

Let $\map {\mathcal R} {\mathbb J}$ be the set of all Riemann integrable functions on $\mathbb J$.

Then $\struct {\map {\mathcal R} {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.