Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. When the bounded linear map $y \mapsto f_y$ of $F$ into $E^*$ the continuous dual of $E$ happens to be an isometric isomorphism, we say that $F$ is the dual of $E$ via $B$. (This can all be found in Dixmier Von Neumann Algebras) Why is it then true that
$||x||=sup_{||y|| \leq 1} |B(x, y)| \forall x \in E$? He states both this and its analog with x, y interchanged, and E, F interchanged. That version is pretty much the definition of what it means to be an isometry, but this one here is not.