Hilbert space properties for PDEs

 In the context of PDEs the following properties are important for the Hilbert spaces.

Notation: Let $V$ be Hilbert space and let $a(.,.):V\times V \rightarrow \mathcal{R}$ be a bilinear form.
Property 1:
For PDE solutions, we need this bilinear form to be bounded.
\begin{equation}
  |a(u,v)| < M||u|| ||v|| \text{ for some $M>0\in \mathcal{R}$ }
\end{equation}
Property 2:
This is called \(V\)-Ellipticity'. The concept of $V$-ellipticity is crucial in establishing the well-posedness (existence, uniqueness, and stability of solutions) of boundary value problems formulated in a variational framework. It guarantees the uniqueness and stability of solutions to the corresponding variational problems. In essence, if the bilinear form derived from a PDE is V-elliptic, then the solution to the variational problem (and hence to the PDE) depends continuously on the data (such as boundary conditions and external forces), ensuring that small changes in input lead to small changes in the output.
Mathematically, it provides a lower bound for the bilinear form.
\begin{equation}
  a(u,v) \ge \alpha ||v||^2 \text{ $\alpha$ is a constant. $v \in V$}
\end{equation}

Simple example
Simplest example of Hilbert space is Euclidean space with euclidean metric. It is not too difficult to verify that the euclidean metric is bilinear, it is bounded in the property $1$ sense and it has $V$-Ellipticity (Property $2$).