Subspace of Real Continuous Functions

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

Let $\mathcal C \left({\mathbb J}\right)$ be the set of all continuous real functions on $\mathbb J$.

Then $\left({\mathcal C \left({\mathbb J}\right), +, \times}\right)_\R$ is a subspace of the $\R$-vector space $\left({\R^{\mathbb J}, +, \times}\right)_\R$.

Proof
By definition, $\mathcal C \left( {\mathbb J} \right) \subseteq \R^{\mathbb J}$.

Let $f, g \in \mathcal C \left({\mathbb J}\right)$.

It needs to be shown that:


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


 * $\left({2}\right): \quad \left({f \times g}\right) \in \mathcal C \left({\mathbb J}\right)$

$\left({1}\right)$ follows by Sum of Continuous Functions is Continuous.

$\left({2}\right)$ follows by Product of Continuous Functions is Continuous.