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

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

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

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

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


 * $\paren 1: \quad f + g \in \map \CC {\mathbb J}$


 * $\paren 2: \quad \lambda f \in \map \CC {\mathbb J}$ for any $\lambda \in \R$

$\paren 1$ follows by Sum Rule for Continuous Functions.

$\paren 2$ follows by Multiple Rule for Continuous Functions.

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