Let $X$ be a normed space and $X′$ its dual space. If $X≠{0}$, show that $X′$ cannot be ${0}$.
Theorem: For an element $x_0$ other than $0$, there is a functional $f^\sim$ such that the norm is $||f||^\sim=1$ and $||f||^\sim=||x_0||$.
I think I will use this theorem. But how do I prove it, can you help me?