Why only densely defined operators can have an adjoint operator?

Question

Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?

Answer 1

The adjoint is defined as

$$ \langle T^\ast u, v\rangle = \langle u, Tv\rangle. $$

This only defines $T^\ast u$ uniquely if the set of all $v$, for which this equation makes sense is dense.

But this means that $T$ has to be densely defined.