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

Question

Thanks in advance.

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.

I appreciate your help.

Answer 1

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.

Answer 2

The paper is referring to distributions.