Consider $\sum_{i,j=1}^n \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial^2 x_i} \overline{\dfrac{\partial^2 v}{\partial^2 x_j} } dx + \lambda \displaystyle\int_{\mathbb{R}^n} u \overline{v} dx$ and with this scalar product we can define a norm which is equivalent to the classical norm on $H^2(\mathbb{R^n})$.
How can use the Riesz theorem to prove that, for all $\lambda > 0,$ and for all $f \in H^{-2}(\mathbb{R^n})$ there exist a unique $u \in H^2(\mathbb{R^n})$ solution of the equation $(\Delta^2 + \lambda) u = f$?
Thanks.
Hint (or solution...): Take the Riesz-representative of $f$.