Consider the following problem: $$ -\nabla^2 u = f,$$
Referencing to this post: FEM, we write the problem in variational form
I'm assuming Dirichlet boundary conditions here): Find $u\in H^1_0(\Omega)$ such that $$ a(u,v):= \int_\Omega \nabla u\cdot \nabla v \,dx = \int_\Omega fv\,dx \qquad\text{for all }v\in H^1_0(\Omega).$$ The important property here is that $$ a(v,v) = \|\nabla v\|_{L^2}^2 \geq c \|v\|_{H^1}^2 \qquad\text{for all }v\in H^1_0(\Omega). \tag{1}$$ (This follows from Poincaré's inequality.)
Now the classical finite element approach is to replace the infinite-dimensional space $H^1_0(\Omega)$ by a finite-dimensional subspace $V_h\subset H^1_0(\Omega)$ and find $u_h\in V_h$ such that $$ a(u_h,v_h):= \int_\Omega \nabla u_h\cdot \nabla v_h \,dx = \int_\Omega fv_h\,dx \qquad\text{for all }v_h\in V_h.\tag{2}$$ The important property here is that you are using the same $a$ and a subspace $V_h\subset H^1_0(\Omega)$ (a conforming discretization); that means that you still have $$ a(v_h,v_h) \geq c \|v_h\|_{H^1}^2 >0 \qquad\text{for all }v_h\in V_h. \tag{3}$$
How can I be sure that $\|v_h\|_{H^1}^2 >0$ ?
