Yes and not, saigmqnthe spaces l2 (with lower case L) consisting on the square integrable functions is the one that we are descrbing. As you sais, there are functions that for which their distance (in the l2 norm) is zero.This defines an equivalence relation in l2 (agan lower case L).By taking the quotient of the space by this equivaence relation we obtain the space L2 (now upper case L), in which the norm is a proper one, that is, the distance of two "object" (they are not functions anymore but quotients) is zero if and only if the onject is the same.however, it's always more intuitive to continue thinking that this objects ar ethe squeare integrable functions, even if we are just working with some representations of the quotient space.i do not if i was very clear here, but it's late.