Subspace of Real Continuous Functions

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

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

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