how take weak derivative of norms in hilbert spaces?

Question

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); u_t∈L^2 ([0,T];V^* )$. I have to show that :

$\frac{d}{dt}$$‖u‖_H^2=2⟨u,u_t⟩$ in the distribution sence on $(0,T)$

And I have no idea how to get from the product in H to ⟨,⟩.

This to me makes no sence since for one $H$ there can be many $V$ for example the sobolev spaces: $H^m_0↪L^2$ so I belive I'm missing some details

The only explanation I have is as follows: When we use the continus dens incusions, we automaticaly assumed that the difference in computation betwin $(*,*)$ and $<*,*>$ is THE DOMAIN of definition not the actual computation

Answer 1

This is calculus in Bochner spaces. Theorem 3 of Chapter 5.9 in Evans' PDE book explains that identity very well for the particular choice $V=H^1_0$ and $H=L^2$; the generalization should be not too difficult.