Bounded linear functionals over smooth maps of a compact interval

Question

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with the family of seminorms $\| \varphi \|_m = \displaystyle\sup_{x \in [0; 1]} \left| \varphi^{(m)}(x) \right|$ :

1. For any continuous linear form $L$ on this space, is there an integer $m \ge 0$ and continuous map $g : [0; 1] \to \mathcal{R}$ such that : $\forall \varphi \in E, L(\varphi) = \displaystyle \int_0^1 g(t) \varphi^{(m)}(t) dt$ ? I know that $L$ can be seen as the restriction of a compactly supported distribution $T \in \mathcal{D}'(\mathbb{R})$ just by restricting the test functions to $[0; 1]$, and that $T$ is a sum of derivatives of order $\le m$ of continuous maps with compact support in $(-\varepsilon; 1+\varepsilon)$ for every $\varepsilon$, but I don't know what means for $L$.

2. If the answer to the first question is yes, then what is the kernel of the map $\Phi_m : g \in \mathcal{C}^0([0; 1]) \mapsto \left( \varphi \in E \mapsto \displaystyle \int_0^1 g(t) \varphi^{(m)}(t) dt \right)$ ? In $\mathcal{D}'(\mathbb{R})$ that would imply that $g$ is a polynomial of degree $\le m$ (I think!), but it obviously doesn't work here.

Thanks in advance for any pointers!

Answer 1

The answer to question 1 is negative. For example, $L(\varphi)=\varphi(0)$ is not of this form. Neither is $L(\varphi)=\varphi'(1/2)$, etc.

A continuous functional on $E$ is bounded by some finite collection of seminorms; therefore it is also a continuous functional on $C^m([0,1])$ for some $m$. The space $C^m([0,1])$ is a direct sum $P_m\oplus V_m$ where $P_m$ is the space of polynomials of degree less than $m$ and $V_m$ is the space of functions $f$ such that $f(0)=f'(0)=\dots=f^{(m-1)}(0)=0$. Thus, the dual of $C^m([0,1])$ is the direct sum of $P_m^*$ and $V_m^*$.

• Every linear functional on $P_m$ is of the form $L(p) = \sum_{k=0}^{m-1} c_k p^{(k)}(0)$.
• Every linear functional on $V_m$ is of the form $L(f) = \int_0^1 f^{(m)}(x)\,d\mu(x)$ for some signed measure $\mu$. This follows from the Riesz representation theorem for $C[0,1]$, since $f\mapsto f^{(m)}$ is an isomorphism between $V_m$ and $C[0,1]$.

Therefore, every linear functional on $C^\infty[0,1]$ is of the form $$L(f) = \sum_{k=0}^{m-1} c_k f^{(k)}(0)+\int_0^1 f^{(m)}(x)\,d\mu(x)$$ for some integer $m\ge 0$, some constants $c_0,\dots,c_{m-1}$, and some signed measure $\mu$.