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 $$\left({\R^{\mathbb J}, +, \times}\right)_\R$$.