Let $C^k(\mathbb{R})$ denote the vector space of $k$-times continuously differentiable functions $\mathbb{R}\to\mathbb{R}$ (with $k\in\mathbb{N}\cup\{0,\infty\}$), and $C^k_c(\mathbb{R})\subset C^k(\mathbb{R})$ denote the subspace of compactly supported functions. Let $D^k(\mathbb{R})$ denote the algebraic dual of $C^k(\mathbb{R})$, i.e. the set of linear maps $C^k(\mathbb{R})\to\mathbb{R}$. $D^k(\mathbb{R})$ contains familiar functionals such as $f\mapsto f(x)$ and $f\mapsto\int\psi f$, but it also contains less familiar objects.
In particuar, let $A\subset D^k(\mathbb{R})$ denote the subspace of functionals which vanish on $C_c^k(\mathbb{R})$, or equivalently the kernel of $\iota^*$, where $\iota:C^k_c(\mathbb{R})\to C^k(\mathbb{R})$ is the inclusion map. It's straightforward enough to show that $A$ is nonempty on algebraic grounds, but nonetheless difficult to describe the set analytically.
My question, then, takes one of two forms:
- Is there an explicit example of a nonzero element $\lambda\in A$, ideally defined in such a way that one could in principle compute $\lambda(f)$ for a simply defined (e.g. polynomial) $f$?
- Alternately, is there a reason (set-theoretic or otherwise) that $A$ does not contain any "easily-constructed" functionals?
There are no "explicit" examples, and indeed it's for set-theoretic reasons. It's consistent with ZF+DC that $A=0$. Hence, you can't construct a nonzero functional in $A$ except by using the axiom of choice in an essential way.
The usual topology on $C^k(\mathbb{R})$ of "uniform convergence of $k$ derivatives on compact sets" is Fréchet; it's generated by the countably many seminorms $|f|_n = \sum_{i=1}^k \sup_{[-n,n]} |f^{(i)}|$, which induce the complete translation-invariant metric $\rho(f,g) = \sum_{n=1}^\infty 2^{-n} (|f-g|_n \wedge 1)$. Note that $C^k_c(\mathbb{R})$ is dense in this topology (cutoff functions). So the only linear functional in $A$ which is continuous in this topology is $0$.
Now it's consistent with ZF+DC that every linear functional on a Fréchet space $X$ is continuous. See for instance Schecter, Handbook of Analysis and its Foundations, Theorem 27.45 (ii). (His "F-spaces" include Fréchet spaces.) The idea is that a discontinuous linear functional would let you construct a subset of $X$ that doesn't have the property of Baire (BP), and Solovay and Shelah constructed models of ZF+DC in which every subset of every Polish space has the BP.