Subspace of Real Continuous Functions

Theorem
Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$.

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

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

Proof
By definition, $\map C {\mathbb J} \subseteq \R^{\mathbb J}$.

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

By Two-Step Vector Subspace Test, it needs to be shown that:


 * $(1): \quad f + g \in \map C {\mathbb J}$


 * $(2): \quad \lambda f \in \map C {\mathbb J}$ for any $\lambda \in \R$

$(1)$ follows by Sum Rule for Continuous Real Functions.

$(2)$ follows by Multiple Rule for Continuous Real Functions.

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