**Step 1: Reduction.** If $v \ne w$, then $v - w \ne 0$, so it suffices to show: for every $u \in V \setminus \{0\}$, there exists $f \in V^*$ with $f(u) \ne 0$. **Step 2: Apply the existence of support functionals.** By the [Existence of Support Functionals](/theorems/881), there exists $f_u \in V^*$ with $f_u(u) = \|u\|_V \ne 0$ (since $u \ne 0$). **Step 3: Conclude.** Setting $f = f_u$ and $u = v - w$, we obtain $f(v) - f(w) = f(v - w) = f_u(v - w) = \|v - w\|_V \ne 0$, so $f(v) \ne f(w)$.