Subspace of Real Continuous Functions

Theorem
Let $$\mathbb{J} = \left\{{x \in \R: a \le x \le b}\right\}$$.

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

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