# 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 Rule for Continuous Functions.

$\left({2}\right)$ follows by Product Rule for Continuous Functions.

$\blacksquare$