It is convenient to denote that particular bra vector is mapped to by < psi |.
2.
We can choose this mapping such that any ket vector | psi > is always mapped to that bra vector, maps it to its squared norm.
3.
You can also think of the ket vectors being the elements of a Hilbert space and the bra vectors as linear functionals that map elements from the Hilbert space to the set of complex numbers.
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