Consider flowing exercise:
Let V be Hilbert space and $A: V\times V \rightarrow \mathbb{R}$ be a symmetric, elliptic (with constant $\alpha_1$) and continuous (with constant $\alpha_2$) bilinear form. Let $b:V\times V$ be a continuous bilinear form with constant $\beta_2$. On the space $X := V \times V$ define the bilinear form $B:X \times X \rightarrow \mathbb{R}$ by
\begin{align*} B((u_1,u_2),(v_1,v_2)) = A(u_1,v_1) + b(u_1,v_2) + A(u_2,v_2) \forall (u_1,u_2),(v_1,v_2) \in X \end{align*}
I was able to show that $B$ is inf-sup stable on $X$.
What i was not able to do yet is following:
Provide a possible choice of a subspace $X_h\subset X$ such that the discrete problem is again inf-sup stable. Similarly provide an example $X_h\subset X$ such that the discrete variational problem is not solvable.
It would be nice to have some examples of such subspaces and an explanation on why that spaces fulfill aboves requirements.