Show the existence of a continuous functional

Question

Let $V$ be a normed space and $v \in V$ be a fixed vector. Show that there exists $\theta \in V^*$ with $\Vert \theta \Vert = 1$ and $\theta(v) = \Vert v \Vert$

I was able to prove this for vectors that are not zero, using Hahn-Banach theorem.

How to handle the case where $v = 0$?

I tried to define a desired functional but was unsuccesful.

Answer 1

As suggested, take any bounded functional $f \in V^*$ such that $\|f\| = 1$.

Then we have $$f(0) = 0 = \|0\|$$