1. The problem statement, all variables and given/known data Claim: The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space". Proof: Assume Lu=0 and Lv=0 (i.e. have two solutions) (i) By linearity, L(u+v)=Lu+Lv=0 (ii) By linearity, L(au)=a(Lu)=(a)(0)=0 => any linear combination of the solutions of a linear homoegenous PDE solves the PDE => it forms a vector space 2. Relevant equations N/A 3. The attempt at a solution and comments Now, I don't understand why ONLY by proving (i) and (ii) alone would lead us to conclude that it is a vector space. There are like TEN properties that we have to prove before we can say that it is a vector sapce, am I not right? Are there any theorem or alternative definition that they have been using? Thanks!