Today brings us to the most important definition in distributional calculus: the distributions themselves.

Here's the formal definition using the set of test functions $\mathcal{D}(\mathbb{R})$ we defined earlier:

Any linear functional $T: \mathcal{D}(\mathbb{R}) \to \mathbb{R}$ a distribution.In addition, for a locally integrable function $f(x):X\to\mathbb{R}$, a corresponding distribution can be defined by$$T_f(\phi)=\int_{\mathbb{R}}f(x)\phi(x)\;dx.$$We usually write $\langle T, \phi\rangle$ instead of $T(\phi)$ and call the set of all distributions of this type $\mathcal{D}'(\mathbb{R})$.

There are only two things needed to truly understand this definition; how to take the average of a continuous function and what test functions are. Check out the integrand. Multiplying the target function $f(x)$ by each individual test function $\phi(x)$ has the effect of scaling $f(x)$ at every point---in particular, the integrand zeros out outside the support of $\phi(x)$, while the other points are weighted depending on $\phi(x)$. Hence every individual component of the definition a weighted average of $f(x)$ over a compact set. (Strichartz directly compares this to finding the temperature of a room with a thermometer: it won't display the temperature at one point, rather the average temperature of some portion of the area.) If each of these weighted averages are known for every existing $\phi(x)$, that is what defines the distribution.

Defining distributions in this way lets us account for objects that we think look like functions, but actually aren't. The Dirac delta function is the perfect example---the infinite value at zero ruins anything, so it isn't really a function*. However, the integral of $\delta(x)$ is bounded no matter what test function we weight it by, so the 'average' exists over every possible range, meaning $\delta(x)$

*is*a distribution. In particular, $$\langle \delta,\phi\rangle = \phi(0).$$

It would be useful to go over a couple useful properties of distributions, starting with the issue of consistency. This was supposed to happen today! Unfortunately, I'm dead tired and need to go lie down forever. Let's leave the important properties for next week.

* The Dirac delta function is to functions what killer whales are to whales... a complete misnomer.

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