My professor just presented to us his proposition, which is as follows: A functional $L\colon V\to R$ (where $V$ is a vector space and $R$ is a field) is linear if and only if:
- $L(v+w)=L(v)+L(w)$;
- $L(av)=aL(v)$ for all $v,w$ in $V$, and all real numbers $a$.
When I try to prove this, I can't help but feel that I am proving the definition of linearity, which is my assumption. Am I making some sort of conceptual mistake?