<Mentor's note: moved from a technical forum, therefore no template.> I'm long out of college and trying to teach myself QM out of Shankar's. I'm trying to understand the reasoning here because I think that I am missing something... 1.1.3 1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space? 2) How about periodic functions? obeying f(0)=f(L) ? 3) How about functions that obey f(0)=4 ? If the functions do not qualify, list what go wrong. The first 2 questions seem straight forward. X and L go to zero, and outside those boundaries you would get the null vector. If the function is periodic, you are certain to return a vector in the f(x). #3 is tripping me up. I thought of an example function... lets say f(x)=x+4, so f(0)=4. How is this not a vector space? I was told that if g(x) and h(x) were in the set of f(x), g(0)+h(0)=8 which is not 4. I do understand how this proves anything? What am I misunderstanding?