Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$.
Does this mean that $M$ is a linear map?
If not, could someone please give me an example?
It seems to me that some continuity condition is needed.
Thank you!