Subspace of Smooth Real Functions

Corollary to Subspace of Real Functions of Differentiability Class
Let $\mathbb J = \set {x \in \R: a < x < b}$ be an open interval of the real number line $\R$.

Let $\map {\CC^{\paren \infty} } {\mathbb J}$ denote the set of all continuous real functions on $\mathbb J$ which are differentiable on $\mathbb J$ at all orders.

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

Proof
Note that by the definition of smooth real function:


 * $\ds \map {\CC^{\paren \infty} } {\mathbb J} = \bigcap_{m = 0}^\infty \map {\CC^{\paren m} } {\mathbb J}$

By Subspace of Real Functions of Differentiability Class:


 * $\struct {\map {\CC^{\paren m} } {\mathbb J}, +, \times}_\R$ is a subspace of $\struct {\R^{\mathbb J}, +, \times}_\R$ for all $m$.

Then, from Set of Linear Subspaces is Closed under Intersection:


 * $\ds \struct {\map {\CC^{\paren \infty} } {\mathbb J}, +, \times}_\R$ is a subspace of $\struct {\R^{\mathbb J}, +, \times}_\R$.