is there any natural way to define a inner product on $V^*$?
First we could consider Riesz isomorphism $\mathfrak{R}:V\rightarrow V^*$, and define $\langle F, G\rangle_{V^*}:=\langle \mathfrak{R}^{-1}(F),\mathfrak{R}^{-1}(G)\rangle_{V}$.
With this definition, if $u,v\in V$ and $\mathfrak{G}:V\rightarrow V^*$ is the injective map defined by the Gelfand triple, will $\langle \mathfrak{G}(u), \mathfrak{G}(v)\rangle_{V^*}=\langle u, v\rangle_{H}$? In other words, does $\langle \mathfrak{R}^{-1}\mathfrak{G}(u), \mathfrak{R}^{-1}\mathfrak{G}(v)\rangle_{V}=\langle u,v\rangle_{H}$ hold for $u,v\in V$?
I know that given $u\in V$, we have $\langle\mathfrak{G}(u), v\rangle_{V^*,V}=\langle u, v\rangle_{H}$ for $v\in V$, but didn't found much information about $\langle\cdot, \cdot\rangle_{V^*}$.
My model case is $H=L^2(\Omega)$, $V=H_0^1(\Omega)$ and $V^*=H^{-1}(\Omega)$. I want to solve some parabolic PDE, so I consider the space $\mathcal{H}=\{u\in L^2(\mathbb{R},V):\: u'\in L^2(\mathbb{R},V^*)\}$ (where $u'$ is defined weakly) and $\mathcal{V}=C_c^{\infty}(\mathbb{R}\times\Omega)\subset \mathcal{H}$. The space $\mathcal{H}$ has a inner product given by $$ \langle u,v\rangle_{\mathcal{H}}=\int_{\mathbb{R}}\langle u(t), v(t)\rangle_V\, dt+\int_{\mathbb{R}}\langle u'(t), v'(t)\rangle_{V^*}\, dt. $$ I am trying to find the right norm to put on $\mathcal{V}$ in order to be able to apply Lions-Lax-Milgram to $B:\mathcal{H}\times\mathcal{V}\rightarrow\mathbb{R}$ given by $$ B(u,\varphi)=-\int_{\mathbb{R}}\langle u(t),\varphi'(t)\rangle_H\, dt +\int_{\mathbb{R}}\langle A(t)\nabla_X u(t),\nabla_X \varphi(t)\rangle_{L^2(\Omega,\mathbb{R}^n)}\, dt, $$ with the right hand side $F(\varphi)=B(h_0,\varphi)$ for fixed $h_0\in\mathcal{H}$. I would like to have on $\mathcal{V}$ something simmilar to $\|\cdot\|_\mathcal{H}$, but I have problems with the coercivity condition. The main question is related to this problem: For $\varphi,\psi\in\mathcal{V}$ we have $\varphi,\psi\in\mathcal{H}$ so $\varphi'(t),\psi'(t)\in V^*$ for each $t$, but we also have $\varphi'(t),\psi'(t)\in V$ (at least for the partial derivative, which should be identified with the weak derivative). So, what does $$ \int_{\mathbb{R}}\langle\varphi'(t),\varphi'(t)\rangle_{V^*}\,dt $$ mean? Should we consider the scalar product in $H$ inside the integrand or the one induced by Riesz isomorphism? Will they coincide?
Thanks for your help