I have a question about inf-sup condition. In these at page 14 is explained the Mixed Formulation in Mixed Finite Elements. We can re-write the problem in this way: let $A:V\to V^\ast, B:V\to Q^\ast, K = \operatorname{Ker} B\subset V, F\in V^\ast,G\in B^\ast $., then $$ \begin{cases} Au + B^Tp = F \\ B u = G \end{cases} $$ In another book it says that the two inf-sup condition for $a$, i.e $$ \begin{split} \inf_{v_0\in K} &\sup_{w_0\in K} \dfrac{a(v_0,w_0)}{\|v_0\|\|w_0\|}>\alpha_1 \\ \inf_{w_0\in K} &\sup_{v_0\in K} \dfrac{a(v_0,w_0)}{\|w_0\|\|v_0\|}>\alpha_2 \end{split} $$
are equivalent to the fact that $A_{KK'}$ is an isomorphism. I try to go more in depth with the details. I can define $A_{KK'}$ given $A$ as a restriction of $A$ to the $K$, i.e $$ A_{KK'}u_0(v_0) = a(u_0,v_0) $$ However I am not able to prove these inf-sup conditions from the fact that $A_{KK'}$ is an isomorphism, while I am able to derive it from the coercivity of $A$ in the Kernel. The book suggests to follow the same argument of the first inf-sup condition for $B$: in this case, the inf-sup condition follows from the surjectivity of $B$. I think that the first one follows from the surjectivity of $A_{KK'}$ similarly, while the second one from the surjectivity of the transpose.
Any other helps?
Thank you!