Subspace of Real Continuous Functions

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

Let $\mathcal C \left({\mathbb J}\right)$ be the set of all continuous real functions on $\mathbb J$.

Then $\left({\mathcal C \left({\mathbb J}\right), +, \times}\right)_\R$ is a subspace of the $\R$-vector space $\left({\R^{\mathbb J}, +, \times}\right)_\R$.