Characterizing germs of smooth functions

Question

There's a sheaf of smooth real-valued functions on $\mathbb{R}$, and its germ at $0$ is some vector space $V$. I would like to understand this space. There is a surjective linear map

$$ \phi \colon V \to \mathbb{R}^{\mathbb{N}} $$

sending the germ $[f]$ of any smooth function $f: \mathbb{R} \to \mathbb{R}$ to its list of derivatives:

$$ \phi([f]) = (f(0), f'(0), f''(0), \dots ) $$

This is the 'easily understandable aspect' of $V$. I want to understand the kernel $ \mathrm{ker}(\phi)$.

So:

Question. Can we explicitly construct any nonzero linear map $\ell : \mathrm{ker}(\phi) \to \mathbb{R}$?

In simple terms: can can we extract any real numbers from the germ of a smooth function $f \colon \mathbb{R} \to \mathbb{R}$ in a linear way, other than by taking derivatives of that function at $0$?

Since $\ker(\phi)$ is infinite-dimensional, there exist infinitely many linearly independent linear maps $\ell : \mathrm{ker}(\phi) \to \mathbb{R}$. But this does not imply that we can actually get our hands on any of them, because it's possible that my last sentence can only be proved using the axiom of choice (or some weaker nonconstructive principle). There are some well-known examples of this frustrating situation in analysis, like the concept of Banach limit.

Someone asked a less precise version of this question on Math Stackexchange:

• What uniquely characterizes the germ of a smooth function?

but it did not receive helpful answers, other than one pointing out that the question originally asked could be phrased in this way. My question is much less ambitious: I'm not asking for enough linear functionals $\ell \colon \mathrm{ker}(\phi) \to \mathbb{R}$ to separate points in $\mathrm{ker}(\phi)$, although that would be great; I'm just asking if any explicit linear functionals of this form are known.

Answer 1

This is a comment (but I am not entitled) since it doesn‘t address the specific question but what I presume to be the underlying one, as hinted in the title, namely what is the appropriate functional analytic structure on the germs of smooth functions at a point. The short answer is that of a complete, conuclear convex bornological space (in the sense of Hogbe Nlend). Before explicating this statement, I will begin by enumerating some rules of thumb on specifying such structures for the standard spaces (of functions, distributions$\dots$):

1. The structure should be complete—if not, one is either looking at the wrong space or wrong topology. Classical example—$C([0,1])$ with the $L^1$ norm. One should extend the function space to $L^1$ or strengthen the topology to the uniform norm.

2. Desirable spaces are determined by growth or smoothness properties, or a combination thereof. There are three possibilities:

3a) There is one such condition—this generally leads to a Banach space(typical examples: bounded continuous, holomorphic or measurable functions, integrable functions $\dots$).

3b) Infinitely many such (often, but not always countably infinite). Then there is a dichotomy:

3c) the conditions come with a universal quantor ($\forall$). This is the typical situation where a locally convex structure is appropriate (prominent example——$C^\infty$-functions).

3d) they come with an existence quantor ($\exists$). Here the appropriate structure is generally that of a convex bornological space (Waelbroeck, Buchwalter, Hogbe Nlend). This is a concept which can be regarded as dual to that of a topolog, where the basic notion is that of a bounded (in contrast to that of an open) set. More formally, locally convex spaces come from the pro category generated by that of Banach spaces, convex bornological spaces from the ind category.

The typical example of this situation is the space of distributions (on a compact interval) or that of tempered distributions.

This suggests that since your space is naturally representable as a union of a sequence of Fréchet spaces, the natural structure is that of a convex bornological space—the bounded sets are those which are so in a $C^\infty$ space on some neighbourhood of zero.

As a final (warning) remark, one major disadvantages of cbs theory compared with that of lcs‘s is the lack of a Hahn-Banach theorem. The dual space can collapse. This happens in a weaker form in your example. The dual space is non-trivial (it consists of the distributions with support at the origin, i.e., the linear combinations of the delta distribution there and its derivatives) but is too small to separate points.