construct Lagrangian for primal we get
$L(w,b,\alpha) = \frac{1}{2}||w||^2 - \sum_1^m\alpha_i[y^{(i)}(w^Tx^{(i)} + b)-1]$
take the derivatives with regard to $w$ and $b$ and plug them back we get:
$L(w,b,\alpha) = \sum_i^m\alpha_i - \frac{1}{2}\sum_{i,j=1}^my^{(i)}y^{(j)}\alpha_i\alpha_j(x^{(i)})^Tx^{(j)}$
with the constraints we finally get the dual form:
$\max_\alpha W(\alpha) = \sum_i^m\alpha_i - \frac{1}{2}\sum_{i,j=1}^my^{(i)}y^{(j)}\alpha_i\alpha_j\langle x^{(i)}x^{(j)}\rangle$
s.t. $\alpha_i\ge0, i = 1,...,m$
$\sum_i^m \alpha_iy^{(i)} = 0$