Some notes on distributions

Formally, distributions are usually defined as continuous linear functions on a certain topological vector space of “test functions”. However, it is not surprising that useful mathematical objects such as distributions also have a natural, concrete description. We shall start from this viewpoint, and later explain the abstraction to topological vector spaces, and explain why the latter is important also. We begin with the example of the most famous distribution: δ0, the Dirac δ-function on R. This is colloquially described as a nonnegative function from R to R with the contradictory properties that it equals zero at all nonzero arguments, but yet that its integral over R equals 1. Of course no such function can exist. Nevertheless, there are two complementary ways to make sense out of δ0. One is to relate it to the linear functional Λ : C(R) → C which sends a function ψ(x) to Λψ := ψ(0). Another is to take a fixed, smooth nonnegative function with rapid decay and integral 1, such as f(x) = e−πx 2 , and study the sequence fT (x) = Tf(xT ) as T → ∞. Each of the functions fT also A perfect example is an element of the Schwartz space S(R), which consists of all smooth functions g on R such that lim|x|→∞ |x|g(x) = 0 for all n, m ≥ 0.