To demonstrate a vector space, we have to show that the vectors (in this case, the functions) with addition form an abelian group and that the scalars (in this case, real numbers) distribute over sums of vectors and vice versa.
Let be given and define . Addition of functions is defined as for all , and scaling of a function by is defined as for all . We want to decide if is a vector space.
It is fairly straightforward to check that is an abelian group (in fact a subgroup of real-valued functions on the real line under addition) since and similarly for . What does this tell us about for nonzero ?
Similarly, the distributive laws fall out of the field structure of but we must be careful that scaled functions are still vanishing at the endpoints. This is, of course, easy to show by a similar argument to the addition closure.