Distribution (functional analysis)

Main idea: usual functions can be reinterpreted like acting over an special set of functions: the test functions (functions with compact support). So we can think of functions like something similar to covectors. But there are more operators than the usual functions: they are the distributions.

Let us define $D (R^{n})$ as the set of $C^{\infty}$ -functions with compact support. We call a distribution to every continuous linear transformation:

T : D (R^{n}) \mapsto R

Usually, the action will be denoted, for $φ \in D (R^{n})$ :

< T, φ >:= T (φ)

Any locally integrable function $f : R \mapsto R$ define a distribution $T_{f}$ in the following way

< T_{f}, φ >= \int_{R^{n}} f (x) φ (x) d x

An important example of distribution, not really coming from a function, is the Dirac delta function $δ$ (not really a function!)

About the nature of distributions as sheafs: see this.

We also can define the derivative of a distribution. It would be convenient that for usual functions $(T_{f})^{'} = T_{f^{'}}$ , that is

< (T_{f})^{'}, φ >= \int_{R^{n}} f^{'} (x) φ (x) d x = [f (x) φ (x)]_{- \infty}^{\infty} - \int_{R^{n}} f φ^{'} d x = - ⟨ f, φ^{'} ⟩

where we have integrated by parts.
So in general we define $T^{'}$ as

⟨ T^{'}, φ ⟩ = - ⟨ T, φ^{'} ⟩

From this point of view it can be check that the Dirac delta is the derivative of the Heaviside function

H (x) = {\begin{cases} 0, & x < 0 \\ \frac{1}{2}, & x = 0 \\ 1, & x > 0 \end{cases}

Attention: value at 0 could be controversial.

At the same way that Dirac delta means evaluation at 0 of a test function, we can interpret that general distributions are "generalized points", or "matter distribution", or "bodies"; and the action over a function is "evaluate the function over that body". If the distribution have compact support I think we can choose any function, not only test functions.