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