1. The problem statement, all variables and given/known data Let V be the set of all nonconstant functions with operations of pointwise addition and scalar multiplication, having the real numbers as their domain. Is V a vectorspace? 2. Relevant equations None. 3. The attempt at a solution My guess is, no. For example F(x) = x2 Among the axioms for vector space, for an arbitrary element of a vector space u there must be a -u. We can put any number in x and it will end up positive. We can multiply it by any negative scalar, but then squaring it will make it positive. Am I right? Am I wrong? Am I right for the wrong reasons? If I am right, there's another disturbing question that comes up: the explanatory portions of the chapter I'm working on state that "The set of all functions having the real numbers as their domain, with operations of pointwise addition and scalar multiplication, is a vector space." Well, isn't the set of all nonconstant functions a subset of that set?