In the book "Mathematical Concepts of Quantum Mechanics" (Gustafson and Sigal) the dual space $V^*$ of a vector space V is defined as the set of bounded linear functionals on $V$. The book goes on introducing a norm on $V^*$:
$$ ||f|| = \sup_{x \in V, ||x||=1} | f(x)|. $$
I don't understand how the evaluation of the sup on the unit sphere of $V$ alone can lead to the condition that $||f||=0 \implies f = 0$, where the $0$ function of $V^*$ would be the function mapping every element of $V$ in the $0$ of the complex plane. I can imagine a function that maps the unit sphere of $V$ into $0$ but maps the rest of $V$ into non-zero elements of $\mathbf{C}$, that then would have a zero norm but would differ from from the null function of $V^*$. Am I missing part of the reasoning?