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In set theory, a function ${f: X \rightarrow Y}$ is defined as an object that evaluates every input ${x}$ to exactly one output ${f(x)}$. However, in various branches of mathematics, it has become convenient to generalise this classical concept of a function to a more abstract one. For instance, in operator algebras, quantum mechanics, or non-commutative geometry, one often replaces commutative algebras of (real or complex-valued) functions on some space ${X}$, such as ${C(X)}$ or ${L^\infty(X)}$, with a more general – and possibly non-commutative – algebra (e.g. a ${C^*}$-algebra or a von Neumann algebra). Elements in this more abstract algebra are no longer definable as functions in the classical sense of assigning a single value ${f(x)}$ to every point ${x \in X}$, but one can still define other operations on these “generalised functions” (e.g. one can multiply or take inner products between two such objects).

Generalisations of functions are also very useful in analysis. In our study of ${L^p}$ spaces, we have already seen one such generalisation, namely the concept of a function defined up to almost everywhere equivalence. Such a function ${f}$ (or more precisely, an equivalence class of classical functions) cannot be evaluated at any given point ${x}$, if that point has measure zero. However, it is still possible to perform algebraic operations on such functions (e.g. multiplying or adding two functions together), and one can also integrate such functions on measurable sets (provided, of course, that the function has some suitable integrability condition). We also know that the ${L^p}$ spaces can usually be described via duality, as the dual space of ${L^{p'}}$ (except in some endpoint cases, namely when ${p=\infty}$, or when ${p=1}$ and the underlying space is not ${\sigma}$-finite).

We have also seen (via the Lebesgue-Radon-Nikodym theorem) that locally integrable functions ${f \in L^1_{\hbox{loc}}({\bf R})}$ on, say, the real line ${{\bf R}}$, can be identified with locally finite absolutely continuous measures ${m_f}$ on the line, by multiplying Lebesgue measure ${m}$ by the function ${f}$. So another way to generalise the concept of a function is to consider arbitrary locally finite Radon measures ${\mu}$ (not necessarily absolutely continuous), such as the Dirac measure ${\delta_0}$. With this concept of “generalised function”, one can still add and subtract two measures ${\mu, \nu}$, and integrate any measure ${\mu}$ against a (bounded) measurable set ${E}$ to obtain a number ${\mu(E)}$, but one cannot evaluate a measure ${\mu}$ (or more precisely, the Radon-Nikodym derivative ${d\mu/dm}$ of that measure) at a single point ${x}$, and one also cannot multiply two measures together to obtain another measure. From the Riesz representation theorem, we also know that the space of (finite) Radon measures can be described via duality, as linear functionals on ${C_c({\bf R})}$.

There is an even larger class of generalised functions that is very useful, particularly in linear PDE, namely the space of distributions, say on a Euclidean space ${{\bf R}^d}$. In contrast to Radon measures ${\mu}$, which can be defined by how they “pair up” against continuous, compactly supported test functions ${f \in C_c({\bf R}^d)}$ to create numbers ${\langle f, \mu \rangle := \int_{{\bf R}^d} f\ d\overline{\mu}}$, a distribution ${\lambda}$ is defined by how it pairs up against a smooth compactly supported function ${f \in C^\infty_c({\bf R}^d)}$ to create a number ${\langle f, \lambda \rangle}$. As the space ${C^\infty_c({\bf R}^d)}$ of smooth compactly supported functions is smaller than (but dense in) the space ${C_c({\bf R}^d)}$ of continuous compactly supported functions (and has a stronger topology), the space of distributions is larger than that of measures. But the space ${C^\infty_c({\bf R}^d)}$ is closed under more operations than ${C_c({\bf R}^d)}$, and in particular is closed under differential operators (with smooth coefficients). Because of this, the space of distributions is similarly closed under such operations; in particular, one can differentiate a distribution and get another distribution, which is something that is not always possible with measures or ${L^p}$ functions. But as measures or functions can be interpreted as distributions, this leads to the notion of a weak derivative for such objects, which makes sense (but only as a distribution) even for functions that are not classically differentiable. Thus the theory of distributions can allow one to rigorously manipulate rough functions “as if” they were smooth, although one must still be careful as some operations on distributions are not well-defined, most notably the operation of multiplying two distributions together. Nevertheless one can use this theory to justify many formal computations involving derivatives, integrals, etc. (including several computations used routinely in physics) that would be difficult to formalise rigorously in a purely classical framework.

If one shrinks the space of distributions slightly, to the space of tempered distributions (which is formed by enlarging dual class ${C^\infty_c({\bf R}^d)}$ to the Schwartz class ${{\mathcal S}({\bf R}^d)}$), then one obtains closure under another important operation, namely the Fourier transform. This allows one to define various Fourier-analytic operations (e.g. pseudodifferential operators) on such distributions.

Of course, at the end of the day, one is usually not all that interested in distributions in their own right, but would like to be able to use them as a tool to study more classical objects, such as smooth functions. Fortunately, one can recover facts about smooth functions from facts about the (far rougher) space of distributions in a number of ways. For instance, if one convolves a distribution with a smooth, compactly supported function, one gets back a smooth function. This is a particularly useful fact in the theory of constant-coefficient linear partial differential equations such as ${Lu=f}$, as it allows one to recover a smooth solution ${u}$ from smooth, compactly supported data ${f}$ by convolving ${f}$ with a specific distribution ${G}$, known as the fundamental solution of ${L}$. We will give some examples of this later in these notes.

It is this unusual and useful combination of both being able to pass from classical functions to generalised functions (e.g. by differentiation) and then back from generalised functions to classical functions (e.g. by convolution) that sets the theory of distributions apart from other competing theories of generalised functions, in particular allowing one to justify many formal calculations in PDE and Fourier analysis rigorously with relatively little additional effort. On the other hand, being defined by linear duality, the theory of distributions becomes somewhat less useful when one moves to more nonlinear problems, such as nonlinear PDE. However, they still serve an important supporting role in such problems as a “ambient space” of functions, inside of which one carves out more useful function spaces, such as Sobolev spaces, which we will discuss in the next set of notes.