---
tag: Q0C1H111EN
kind: statement
role: theorem
name: Schwartz impossibility theorem
msc: [46F10, 46F30]
uses: [0KPHENZ5S7]
sources:
  - url: https://en.wikipedia.org/wiki/Distribution_(mathematical_analysis)
---

## Statement

Let $\mathcal{D}'(\R)$ be the space of distributions on $\R$, the dual of the space of test functions.
Let $C(\R)$ be the space of continuous functions, viewed as distributions.
There is no associative algebra $(\mathcal{A}, \circ)$ over $\R$ with these properties:

1. $\mathcal{D}'(\R)$ is a linear subspace of $\mathcal{A}$, and the constant function $1$ is the unit of $\mathcal{A}$.
2. There is a linear map $D \colon \mathcal{A} \to \mathcal{A}$ with $D(u \circ v) = (Du) \circ v + u \circ (Dv)$ for all $u, v \in \mathcal{A}$.
3. The restriction of $D$ to $\mathcal{D}'(\R)$ is the derivative of distributions.
4. The restriction of $\circ$ to $C(\R) \times C(\R)$ is the pointwise product of functions.

## Remarks

The proof computes inside $\mathcal{A}$ with continuous functions such as $x$, $x \log\abs{x} - x$ and their derivatives in $\mathcal{D}'(\R)$.
The Leibniz rule and the pointwise product give identities for products like $x \circ \delta$, where $\delta$ is the [[Dirac delta|0KPHENZ5S7]].
Associativity then leads to a contradiction such as $\delta = 0$.

So distributions cannot be multiplied in a way that keeps all the rules of calculus.
[[Colombeau algebras|XGQT4WEVGG]] keep clauses 1, 2 and 3 and weaken clause 4: they keep the pointwise product of smooth functions only.

## Explanations

### L1 human

Distributions such as the delta function are very useful, but one cannot multiply them freely.
This theorem shows that the problem is not a lack of cleverness.
Any product that extends the product of continuous functions and obeys the product rule for derivatives must break down.

### L2 human

Nonlinear equations with distributional data, for example in shock waves or in quantum field theory, need products of distributions.
The theorem shows which rule must be given up.
In the Colombeau algebra the product agrees with the usual one on smooth functions, and on continuous functions only up to association, a weak equality tested against test functions.