I do not understand how the antilinear correspondence arises between ket vectors in V and bra vectors in V' (dual space):
So I want to know why
$$ \alpha\lt f_1 | + \beta\lt f_2 | $$ corresponds to $$ \alpha^* | f_1 \gt + \beta^* |f_2 \gt $$
where the * represents complex conjugate.
Is there a proof for this?