Versions of Trace Theorems

Question

I have a quick question about the Trace Theorem. I have been using Evans book of Partial Differential Equations to study Sobolev Spaces. The Trace Theorem is given as "If $U$ is bounded and $\partial U$ is $C^{1}$ then there exists a bounded linear functional $T: W^{1,p}(U) \rightarrow L^{p}(\partial U)$". In a different book the Trace Theorem is given such that there is a trace operator $u \mapsto u |_{\Gamma}$ maps $W^{1,p}(\Omega)$ into $L^{p^{\text#}}(\Gamma)$. Where $p^{\text #} := \{ \frac{np-p}{n-p} \text{ for } p=n ; \text{ an arbitrary large constant for $p=n$}; +\infty \text{ for } p > n \}$. How do these two Theorems coincide? What is the standard theorem?