Reference request for the space $C^k[a,b]$

Question

I am looking for some notes/references that discuss basic properties of the space $C^k[a,b]$ of $k$-times differentiable $\mathbb{C}$-valued functions on an interval $[a,b]$.

I am not very familiar with this space, and the naive questions I would like to be able to answer are:

1. Does $C^k$ here refer to the one-sided derivative? Or does it refer to functions on $[a,b]$ that can be extended to a $C^k$ function on a slightly larger open interval $(a-\varepsilon,b+\varepsilon)$?
2. Is $C^k[a,b]$ a Banach $*$-algebra with respect to the norm$$\|f\|_k=\sum_{i=0}^k\left\|\frac{d^i f}{dx^i}\right\|_\infty?$$ (Again, with derivative suitably interpreted.)

Of course, answers to these questions are also welcome.

Answer 1

The definition of $C^k[a,b]$ enables an intuitive understanding of notions defined on open sets. In other words, something true for $C^k\mathbb{R}$ will remain true for $C^k[a,b]$.

A possible definition is:

$C^k[a,b]$ is the space of functions over $[a,b]$ that can be extended by $\tilde f$ in an open set $O$ with $[a,b] \subset O$, such that $\tilde f$ has a continuous $k$-th derivative on $O$.

1. If this definition is satisfied by $f$, then $f^{(k)}$ and the semi-derivatives up to the $k$-th order are equal over $(a,b)$; Over the boundaries the derivatives at $a$ and $b$ can be understood as the corresponding semi-derivative. (If you are not convinced, work with $\tilde f$ over the larger open set and apply the "restriction" to $[a,b]$).
2. I am unsure what is meant by $*$-algebra here. But yes, with this norm it is a Banach space. The proof is a bit technical and lengthy but not very hard, an exemple here.