# what is $C^{-\infty}(\mathbb{R})$

what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way explicitly.

Refer to www-math.mit.edu/~dav/venice.pdf (end of page 2) for where I have seen it.

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Could you please give some context? Where have you seen this notation? – Fly by Night Mar 15 '13 at 19:11
$C^{∞}(\mathbb{R})$ means the function is infinitely differentiable... so maybe this means that it is nowhere differentiable? – Squirtle Mar 15 '13 at 21:15
Um, in the link, it says that $C^{-\infty}(X)$ is the set of distributions on $X$, where $X$ is a manifold. Could you perhaps clarify (or re-frame) your question? – Jesse Madnick Mar 16 '13 at 2:35
@JesseMadnick: but these are distributions in the sense of analysis, not distributions in the sense of differential geometry (subbundles of a tangent bundle) – Colin McQuillan Mar 16 '13 at 2:36
@AlgRev The dual vector space to test functions (usually compactly supported smooth functions): see en.wikipedia.org/wiki/Distribution_(mathematics) – Neal Mar 16 '13 at 4:28

This question has been answered but the enquirer might be interested in more background information. To my knowledge, the notation $C^{-\infty}(\text{R})$ for the distributions was introduced by J. Sebastiao e Silva in his elementary approach to distributions. The idea was to extend the decreasing chain $C^n(I)$ of $n$-times continuously differentiable functions on the compact interval $I$ by introducing the spaces $C^{-n}(I)$, the latter being the distributions on $I$ which are $n$-th derivatives of continuous functions (the dirac delta-function is then in $C^{-2}$). The series $C^{n}$ now runs over all of the integers and one closes it on the left by $C^\infty$, the space of infinitely smooth functions, and on the right by $C^{-\infty}$, the space of all distributions. Corresponding spaces of functions and distributions on the line are defined by standard localisation methods (in the language of category theory by taking projective limits). An introduction to this theory can be found in the book "Elementary Theory of Distributions" by Campos Ferreira.

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The paper is referring to distributions.

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