1. The problem statement, all variables and given/known data Prove that the sum of any two positive operators on V is positive. 2. Relevant equations 3. The attempt at a solution This problem seems pretty simple. But I could be wrong. Should I name two positive operators T and X such that T=SS* and X=AA*? I have a bad history of seeing a proof and thinking "wow, this'll be cake" which leads to lazy thinking. Should I just choose two vectors v1 and v2 and establish two inner products, <SS*v1, v2>+<AA*v1, v2>? I know that A* is real if X is positive, same goes for S. So <AA*v1, v2>=<Av1, Av2>=<A^2v1, v2>. If I'm right here, its easy street.