What does $C_b$ norm mean

Question

Recently I have been reading a paper where the norm $||f||_{C_b}$ and $||f||_{C_b^m}$ appear without definition. So I would like to know what is the default definition in the mathematics community for these two norm? Thanks!

Answer 1

\begin{align*} \|f\|_{C_{b}(K)}=\sup\{|f(x)|: x\in K\}, \end{align*} and \begin{align*} \|f\|_{C_{b}^{m}(K)}=\sum_{k=0}^{m}\|f^{(k)}\|_{C_{b}(K)}, \end{align*} where $K$ is a compact set, here we assume the context is in one-dimensional, for higher dimensional is defined similarly, using the multi-indices $\alpha=(\alpha_{1},...,\alpha_{n})$ that $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}=k$, $0\leq k\leq m$, and consider $\|\partial^{\alpha}f\|_{C_{b}(K)}$.