put $ D = \{ \lambda \in \mathbb{C} : \mid\lambda\mid < 1\}$ and
$A( D ) = \{ f \in C (\overline{D}) : f \vert_{D} \,\text{ is analytic}\} $.
show that $A(D)$ is a closed sub algebra of $C (\overline{D})$ , then it is a with identity?