Subspace of Real Functions of Differentiability Class

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

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

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

Corollary
Let $$\mathcal C^{\left({\infty}\right)} \left({\mathbb J}\right)$$ be the set of all continuous real functions on $$\mathbb J$$ which are differentiable on $$\mathbb J$$ at all orders.

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