In Appendix A.3 of Evans's PDE(2rd edition), I find the definition of $C^k(\bar{U})$ is : $$ C^k(\bar{U}) = \{u\in C^k(\bar{U}) | D^{\alpha}u \text{ is uniformly continous on bounded subsets of U, for all}\,\, |\alpha|\leq k\} $$ where $U\subset \mathbf R^n$ is open.
However, in Adams's sobolev spaces Para 1.28, I find that he said ''' We define the vector spaces $C^m(\bar{\Omega})$ to consist of all those $\phi\in C^{m}(\Omega)$ for which $D^{\alpha}(\phi)$ is bounded and uniformly continous on $\Omega$ for $0\leq |\alpha| \leq m$. ''' where $\Omega$ is a domain.
Are these two definitions equavalent? And why there are different definitons of $C^{m}(\bar{U})$?