Solution
= Solution
$X^*$ is the Banach space of bounded linear functionals. For $T:X\to Y$, define $T^*:Y^*\to X^*$ by $T^*f=f\circ T$. It is linear and $\|T^*f\|\le\|T\|\|f\|$, so bounded (indeed $\|T^*\|=\|T\|$).
= Solution
$X^*$ is the Banach space of bounded linear functionals. For $T:X\to Y$, define $T^*:Y^*\to X^*$ by $T^*f=f\circ T$. It is linear and $\|T^*f\|\le\|T\|\|f\|$, so bounded (indeed $\|T^*\|=\|T\|$).