I don't think this is exactly what you are asking, but here's one thing that comes to mind. One could take the Hilbert space $H_{0}^{1}(\Omega)$ (the closure of $C_{c}^{\infty}(\Omega)$ in the Sobolev $H^{1}$ topology), where say $\Omega\subset\mathbb{R}^{n}$ is a bounded open set with smooth boundary. There are two equivalent inner products on this space. That is, if one defines $$||u||_{1}:=\int_{\Omega}(u^{2}+|\nabla u|^{2})$$ and $$||u||_{2}:=\int_{\Omega}|\nabla u|^{2}$$ then by a Sobolov inequality (sometimes called Poincaré's inequality) we obtain two equivalent norms on $H^{1}_{0}(\Omega)$ arising from inner products - $\int_{\Omega}(uv+\nabla u\cdot\nabla v)$ and $\int_{\Omega}\nabla u\cdot\nabla v$ resp. This type of Sobolev inequality is used extensively by those studying PDEs. If you would like to learn more about this, 1) I'm sure wikipedia offers some basic resources, and 2) there is a decent introductory book by Evans (Partial Differential Equations) - chapter 5. I apologize if this is not what you were looking for and/or if you were already familiar with this.

I don't think this is exactly what you are asking, but here's one thing that comes to mind. One could take the Hilbert space $H_0^1(\Omega)$ (the closure of $C_c^\infty(\Omega)$ in the Sobolev $H^1$ topology), where say $\Omega \subset \mathbb{R}^n$ is a bounded open set with smooth boundary. There are two equivalent inner products on this space. That is, if one defines

$$ \|u\|_1 := \int_\Omega (u^{2}+|\nabla u|^{2}) $$ and $$ \|u\|_2 := \int_\Omega |\nabla u|^{2} $$

Then by a Sobolov inequality (sometimes called Poincaré's inequality) we obtain two equivalent norms on $H^1_0(\Omega)$ arising from inner products − $\int_\Omega(uv+\nabla u\cdot\nabla v)$ and $\int_\Omega \nabla u\cdot\nabla v$ resp. This type of Sobolev inequality is used extensively by those studying PDEs. If you would like to learn more about this, 1) I'm sure wikipedia offers some basic resources, and 2) there is a decent introductory book by Evans (Partial Differential Equations) - chapter 5. I apologize if this is not what you were looking for and/or if you were already familiar with this.

I don't think this is exactly what you are asking, but here's one thing that comes to mind. One could take the Hilbert space $H_{0}^{1}(\Omega)$ (the closure of $C_{c}^{\infty}(\Omega)$ in the Sobolev $H^{1}$ topology), where say $\Omega\subset\mathbb{R}^{n}$ is a bounded open set with smooth boundary. There are two equivalent inner products on this space. That is, if one defines $$||u||_{1}:=\int_{\Omega}(u^{2}+|\nabla u|^{2})$$ and $$||u||_{2}:=\int_{\Omega}|\nabla u|^{2}$$ then by a Sobolov inequality (sometimes called Poincaré's inequality) we obtain two equivalent norms on $H^{1}_{0}(\Omega)$ arising from inner products - $\int_{\Omega}(uv+\nabla u\cdot\nabla v)$ and $\int_{\Omega}\nabla u\cdot\nabla v$ resp. This type of Sobolev inequality is used extensively by those studying PDEs. If you would like to learn more about this, 1) I'm sure wikipedia offers some basic resources, and 2) there is a decent introductory book by Evans (Partial Differential Equations) - chapter 5. I apologize if this is not what you were looking for and/or if you were already familiar with this.

I don't think this is exactly what you are asking, but here's one thing that comes to mind. One could take the Hilbert space $H_{0}^{1}(\Omega)$ (the closure of $C_{c}^{\infty}(\Omega)$ in the Sobolev $H^{1}$ topology), where say $\Omega\subset\mathbb{R}^{n}$ is a bounded open set with smooth boundary. There are two equivalent inner products on this space. That is, if one defines $$||u||_{1}:=\int_{\Omega}(u^{2}+|\nabla u|^{2})$$ and $$||u||_{2}:=\int_{\Omega}|\nabla u|^{2}$$ then by a Sobolov inequality (sometimes called Poincaré's inequality) we obtain two equivalent norms on $H^{1}_{0}(\Omega)$ arising from inner products - $\int_{\Omega}(uv+\nabla u\cdot\nabla v)$ and $\int_{\Omega}\nabla u\cdot\nabla v$ resp. This type of Sobolev inequality is used extensively by those studying PDEs. If you would like to learn more about this, 1) I'm sure wikipedia offers some basic resources, and 2) there is a decent introductory book by Evans (Partial Differential Equations) - chapter 5. I apologize if this is not what you were looking for and/or if you were already familiar with this.