Let $V = C^{\infty}(\mathbb{R}) = \{f:\mathbb{R} \rightarrow \mathbb{R}/ f \text{ is infinitely differentiable}\}$.
Considering $\,W = \{f ∈ V: f(42) = 0, f(\pi) = 0 \}$.
I want to prove that $\,\dim(C^{\infty}(\mathbb{R})/W) < \infty$. I am not sure how to proceed. My idea is to find a basis for function space $V$. Then, find a basis for $W$ and show there is a rest of $n$ vectors which can form a basis of the quotient space, so its dimension is finite. However, I do not know how to find those bases. Any suggestions would be helpful.
Let $u(x)=x-42, v(x)=x-\pi$. Then $u,v$ are $C^{\infty}$ functions. If $f$ is any $C^{\infty}$ function then $f+W=a(u+W)+b(v+W)$ for some real numbers $a$ and $b$: To see this we only need to show that $f-au-bv$ belongs to $W$, which means it vanishes at $42 $ and $\pi$. I will let you find $a$ and $b$ such that this holds. Once this is done it follows that $\,\dim (V/W) \leq 2$.