Dual space of $\mathcal{C}^n [a,b]$.

I just started reading a few days ago about Banach algebras using the Kaniuth's book. In this, it is said that the space $\mathcal{C}^n [a,b]$ of $n$-times continuously differentiable functions is a Banach algebra with the norm $$||f|| := \sum_{k=0}^n \frac{1}{k!} ||f^{(k)}||_\infty, \quad \forall f \in \mathcal{C}^n [a,b].$$ So I have a question, what is the (topological) dual of this space? It is simple curiosity, maybe it is isomorphic to a known space, or to the direct sum of known spaces, I don't have idea.

Solution 1:

There are many ways to describe functionals on $C^n[a,b]$.

Consider the map $$C^n[a,b]\ni f \mapsto \mathbb{R}^n \times C[a,b]$$

$$f \mapsto (f(a), f'(a), \ldots, f^{n-1}(a), f^{(n)})$$

Fact: the map above is a linear isomorphism ( see Taylor formula with integral remainder.

So now you have to find the linear functionals of the space on the right. Using Riesz theorem, we conclude that every linear functional on $C^n[a,b]$ is given by

$$f \mapsto \sum_{k=0}^{n-1} c_k f^{(k)}(a) + \int_{[0,1]} f^{(n)}(t) d \mu(t)$$ where $c=(c_0, \ldots c_{n-1}) \in \mathbb{R}^n$ and $\mu$ is Borel (signed) measure on $[0,1]$. $c$ and $\mu$ are uniquely determined.

Note: This seems a bit strange, since we could have chosen $b$ instead of $a$. And what about the functional $$f \mapsto \int_{[0,1]} f(t) d \nu(t)$$ Or the functionals $$f \mapsto \int_{[0,1]} f^{(k)}(t) d \nu_k(t)$$ for some $1 \le k \le n-1$. Where do they fit in?

That is a very pertinent question. I will respond with this: we have an integration by parts formula that magically takes care of them all, if applied rigorously.

Here is another way to describe functionals on $C^n[a,b]$. Take $n$ points $a_0$, $\ldots$, $a_{n-1}$ in $[a,b]$. Again, we have a linear isomorphism $$f \mapsto (f(a_i), f^{(n)})$$ from $C^n[a,b]$ to $\mathbb{R}^{n}\times C[a,b]$, so we get a different description of linear functionals.

$bf{Added:}$ Another way to describe functionals.

Consider $\phi_0(\cdot)$, $\phi_1(\cdot)$, $\ldots$, $\phi_{n-1}(\cdot)$ in $C[a,b]$. The linear map

$$f \mapsto (f(a), f'(a), \ldots, f^{(n-1)}(a), f^n + \phi_{n-1} f^{(n-1)} + \cdots + \phi_0 f\,)$$ from $C^n[a,b]$ to $\mathbb{R}^n \times C[a,b]$ is linear, bijective (ODE theory) and continuous, so an isomorphism of Banach spaces. Now, representing the evaluation functional $$f \mapsto f(x)$$ using the RHS data involves some linear functionals, that "solves" the ODE.