Subspace of Real Differentiable Functions

Theorem
Let $\mathbb J$ be an open interval of the real number line $\R$.

Let $\map \DD {\mathbb J}$ be the set of all differentiable real functions on $\mathbb J$.

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

Proof
Note that by definition, $\map \DD {\mathbb J} \subseteq \R^{\mathbb J}$.

Let $f, g \in \map \DD {\mathbb J}$.

Let $\lambda \in \R$.

From Linear Combination of Derivatives, we have that:


 * $f + \lambda g$ is differentiable on $\mathbb J$.

That is:


 * $f + \lambda g \in \map \DD {\mathbb J}$

So, by One-Step Vector Subspace Test:


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