Subspace of Real Functions of a Differentiability Class

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Theorem

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

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


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


Corollary

Let $\mathcal C^{\left({\infty}\right)} \left({\mathbb J}\right)$ be the set of all continuous real functions on $\mathbb J$ which are differentiable on $\mathbb J$ at all orders.


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


Proof


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