Editing Colombeau algebra (section)

== Schwartz' impossibility result ==

Attempting to embed the space <math>\mathcal{D}'(\mathbb{R})</math> of distributions on <math>\mathbb{R}</math> into an associative algebra <math>(A(\mathbb{R}), \circ, +)</math>, the following requirements seem to be natural:

# <math>\mathcal{D}'(\mathbb{R})</math> is linearly embedded into <math>A(\mathbb{R})</math> such that the constant function <math>1</math> becomes the unity in <math>A(\mathbb{R})</math>,
# There is a [[partial derivative]] operator <math>\partial</math> on <math>A(\mathbb{R})</math> which is linear and satisfies the Leibniz rule,
# the restriction of <math>\partial</math> to <math>\mathcal{D}'(\mathbb{R})</math> coincides with the usual partial derivative,
# the restriction of <math>\circ</math> to <math>C(\mathbb{R}) \times C(\mathbb{R})</math> coincides with the pointwise product.

However, L. Schwartz' result<ref>L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", ''Comptes Rendus de L'Académie des Sciences'' 239, pp. 847–848 [http://gallica.bnf.fr/ark:/12148/bpt6k3191m/f847.image.langFR]</ref> implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces <math>C(\mathbb{R})</math> by <math>C^k(\mathbb{R})</math>, the space of <math>k</math> times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta.

Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with <math>C(\mathbb{R}) \times C(\mathbb{R})</math> replaced by <math>C^\infty(\mathbb{R}) \times C^\infty(\mathbb{R})</math>, i.e., they preserve the product of smooth (infinitely differentiable) functions only.