I am studying the condition number of the stiffness matrix and I am facing some problems with the proof of the inverse inequality:
$$|| \nabla v_h ||_0^2 \le C h^{-2} || v_h ||_0^2.$$
Using the fact that the triangulation is quasi-uniform and working on the reference element, the author writes that the function
$$ \frac{\int_{\hat{K}} | \nabla \hat{v}|^2}{\int_{\hat{K}} \hat{v}^2} $$
is bounded and therefore we have that
$$ (1) \int_K | \nabla v_h|^2 \le C||B_K^{-1}||^2 \int_K v_h^2 \le \frac{C}{\rho_K^2} \int_K v_h^2. $$
I am not able to understand how to get $(1)$. He says he uses some semi-norm transformation but I cannot understand. Do you have any reference for this?