Let V be the set of all functions
f: $\mathbb{R}$ $\to$ $\mathbb{R}$
such that the n-th derivative "$f^n$" of $f$ exists for all integers $n$.
We define a non standard addition ⊕ on V by setting $f$ ⊕ $g$ = $f'$ + $g'$ where $f'$ and $g'$ are the derivatives of $f$ and $g$ respectively.
V together with non-standard additon, scalar multiplication and $0$ is not a vector space.
Which vector space axioms does V not satisfy?
Assuming the multiplication with scalar is defined in the obvious way. We would need the identity $(\lambda+\mu)f = \lambda f \oplus\mu f$, but in general $$\lambda f \oplus \mu f = \lambda f' + \mu f' \neq \lambda f + \mu f. $$