Let $D$ be the open unit disc in the complex plane and $U=D/\{\frac{-1}{2},\frac{1}{2}\}.$ Also let $H_{1}=\{f:D\rightarrow\mathbb{C} \mid f\text{ is holomorphic and bounded}\}$ and $H_{2}=\{f:U\rightarrow\mathbb{C}\mid f\text{ is holomorphic and bounded}\}.$ Then the map $r:H_1\rightarrow H_2$ given by $r(f)=f|_U$, the restriction of $f$ to $U,$ is
$A.$ Injective but surjective.
$B.$ Surjective but not injective.
$C.$ Both injctive and surjective.
$D.$ Neither injective nor surjective.
According to me it is both injective and surjectve . As for injective $r(f)=r(g)\Rightarrow f|_U=g|_U\Rightarrow f=g$ by identity theorem . For surjective take preimage of $f$ as $f$ itself as $f$ can have atmost removable singularities at $\frac{\pm 1}{2}.$ Please suggest me. Thanks in advance.