For $j=1, 2$, let $V_j \subset H_j$ be a dense and continuous embedding with $V_j$ a Banach space and $H_j$ a Hilbert space. Identify $H_1 = H_1^*$ and $H_2 = H_2^*$ (using Riesz representation) so that we have two Gelfand triplets $$V_1 \subset H_1\subset V_1^*$$ $$V_2 \subset H_2\subset V_2^*.$$
Suppose I have linear continuous bijective maps $F_V:V_1 \to V_2$ and $F_H:H_1 \to H_2$ and $F_V':V_2^* \to V_1^*$.
I wish to use the formula $$\langle a, b \rangle_{V_j^*, V_j} = (a,b)_{H_j}\quad\text{if $a \in H_j$}$$ where $a$ and $b$ may involve the maps above.
Is there any danger in this situation (with the Gelfand triple) given that the spaces are related via the maps $F$ and $G$? AFAIK, the only issue is that if $V_j$ are Hilbert spaces then one should not identify their duals (which is obvious) given that we already identified $H_j$ with their duals. But in my case can be there another problem that I might be missing?
I have read of something called "Gelfand triple isomorphism" but the theory does not seem to go much further than just the definition.
