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  "body": "\n## Statement\n\nLet $\\mathcal{D}'(\\R)$ be the space of distributions on $\\R$, the dual of the space of test functions.\nLet $C(\\R)$ be the space of continuous functions, viewed as distributions.\nThere is no associative algebra $(\\mathcal{A}, \\circ)$ over $\\R$ with these properties:\n\n1. $\\mathcal{D}'(\\R)$ is a linear subspace of $\\mathcal{A}$, and the constant function $1$ is the unit of $\\mathcal{A}$.\n2. 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 $u, v \\in \\mathcal{A}$.\n3. The restriction of $D$ to $\\mathcal{D}'(\\R)$ is the derivative of distributions.\n4. The restriction of $\\circ$ to $C(\\R) \\times C(\\R)$ is the pointwise product of functions.\n\n## Remarks\n\nThe proof computes inside $\\mathcal{A}$ with continuous functions such as $x$, $x \\log\\abs{x} - x$ and their derivatives in $\\mathcal{D}'(\\R)$.\nThe Leibniz rule and the pointwise product give identities for products like $x \\circ \\delta$, where $\\delta$ is the [[Dirac delta|0KPHENZ5S7]].\nAssociativity then leads to a contradiction such as $\\delta = 0$.\n\nSo distributions cannot be multiplied in a way that keeps all the rules of calculus.\n[[Colombeau algebras|XGQT4WEVGG]] keep clauses 1, 2 and 3 and weaken clause 4: they keep the pointwise product of smooth functions only.\n\n## Explanations\n\n### L1 human\n\nDistributions such as the delta function are very useful, but one cannot multiply them freely.\nThis theorem shows that the problem is not a lack of cleverness.\nAny product that extends the product of continuous functions and obeys the product rule for derivatives must break down.\n\n### L2 human\n\nNonlinear equations with distributional data, for example in shock waves or in quantum field theory, need products of distributions.\nThe theorem shows which rule must be given up.\nIn 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.\n",
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