Why the dual norm of a normed vector space is defined on the supremum on $||x||\leq 1$?

Question

From a book I'm reading the definition of the dual norm of a n.v.s. $(X,||\cdot||)$ is defined as

$||f||_{X^*}:=\sup_{||x||\leq 1}|f(x)|, x\in X$ where $f\in X^*$

So why is it not defined like $||f||_{X^*}:=\sup_{||x||= 1}|f(x)|, x\in X$? Is there any instance that those two definitions are not equivalent?