Let $C^{\infty}(I)$ denote the vector space of smooth functions from an interval $I$ to $\mathbb{R}$. Let $\sim$ be the following equivalence relation on $C^{\infty}(I)$: $$ f \sim g \Leftrightarrow f = -g\,.$$
Question: is the quotient $C^{\infty}(I)/{\sim}$ a vector space? What is its dimension?
Your relation is not an equivalence relation, as $f \sim f$ is not true for $f \not \equiv 0$. If you amend your definition to
$$ f \sim g \iff f=g \text { or } f=-g$$
then your construct is not a vector space, as addition is not preserved under the quotient. Take any $f \in C^\infty (I)$ that is not identical to the zero function. We have $f \sim f$ and $f \sim -f$, but not $2f=f+f \sim f+ (-f) = 0$.