Show that $A^o$ is convex, balanced, closed in $A'$

Question

Let $E$ is a normed space and $A \subset E$. Define $$A^o := \{y \in A': |y(x)| \leq 1, \forall x \in A\}.$$ in which, $A'$ is the dual space of $A$.

With this definition, how to show that $A^o$ is convex, balanced, closed in $A'$?