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 $\R^{\mathbb J}$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.5$