Do smooth functions form a convex set?

Question

Consider the set $\mathcal{S}$ of all functions $f: \mathbb{R}^n \rightarrow \mathbb{R}$. It forms a vector space over $\mathbb{R}$. Consider the subset $C^p(\mathbb{R}^n) \subset \mathcal{S}$ of $p$-continuously differentiable functions on $\mathbb{R}^n$ for $p \geq 1$. Is $C^p(\mathbb{R}^n)$ a convex set? That is, for all $t \in [0,1]$ and for any two $f,g \in C^p(\mathbb{R}^n)$, is $tf + (1-t)g \in C^p(\mathbb{R}^n)$?

I am specifically interested in the case $p = 2$.

Answer 1

$C^p(\Bbb R^n)$ is a vector subspace of $\Bbb R^{\Bbb R^n}$.