should any body explain that how do we calculate these step? \begin{align*} L(x,y) &= f(x) + y^T(Ax-b)\\ g(y) &= \inf_x \, L(x,y) \\ &= \inf_x \, f(x) - \langle -A^Ty, x \rangle - \langle y, b \rangle \\ &= - \sup_x \, \langle -A^Ty, x \rangle - f(x) + \langle y, b \rangle \\ &= - f^*(-A^T y) - \langle y, b \rangle. \end{align*} \begin{align*}how\ does\ A\ become\ A^T\ ?\end{align*}
especially step 3, 4,and 5. kindly explain how do i derive each step in details?
thanks