Theorem: Let $X$ be a normed space and let $x_0\neq 0$ be any element of $X$. Then there exists a bounded linear functional $f^\sim$ on $X$ such that $||f^\sim||=1$, $f^\sim(x_0)=||x_0||$.
NOTE: I know the proof of this theorem.
Show that under the assumptions of Theorem there is a bounded linear functional $f^\wedge$ on $X$ such that $||f^\wedge||=||x_0||^{-1}$ and $f^\wedge(x_0)=1$.