Since we can identify functions with vectors for the dual space, why couldn't we also use a regular norm as opposed to the dual norm?

Question

The dual space of a vector space $\mathbb{V}$ is the set of linear functionals $L: \mathbb{V} \to \mathbb{R}.$

By certain important theorem, since every linear functional has the form $L(x) = \langle y, x \rangle$ where $\langle \cdot, \cdot \rangle$ is the ``duality pairing" (same notation as the inner product, ugh), therefore every $L$ could be identified with a vector $y$.

So why go through the trouble of constructing this so-called dual norm, when we can use a regular norm on $y$, for instance, $\|y\|_2$.