Constructing quantum mechanics in a topological vector space endowed with a sesquilinear form Generally it is said that quantum mechanics can be constructed in a topological vector space $V$ and its dual $V^*$, endowed with a sesquilinear form. 
The topology is required for convergence of a set of vectors $\{\psi_n\}_{n\in \mathbb{R}}$, and the sesquilinear form is required for defining  transition amplitude 
$$(\psi,\phi) , \phi \in V , \psi \in V^*.$$ 
Now I want to define adjoint (dual or conjugate) operator for a linear operator $A$ with domain $D(A)\subseteq V$.  However, as I've seen, this operator is defined over Hilbert space, Banach space, and the weaker one, locally convex topological vector space with strong dual. 
So my question is basically that if the space $V$ above is indeed sufficient for defining adjoint operator?
If not, what are the weakest conditions on the vector space that allow defining adjoint operator?
Any comment or suggesting reference would be appreciated. 
 A: There are some strong motivations, in my opinion, to set a quantum theory on Hilbert spaces (natural framework for the representation of involutive algebras of observables).
Nonetheless, the adjoint, or transposed, of a continuous linear map from a top vec space $V$ to a top vec space $W$ can be defined given any $\mathcal{V}$ and $\mathcal{W}$ in compatible duality with $V$ and $W$ respectively (with sesquilinear forms $\sigma_V$ and $\sigma_W$, antilinear in the first argument, that separate points in all spaces). The transpose of a continuous map $u:W\to V$ is a map $^{\mathrm{t}}u:\mathcal{V}\to \mathcal{W}$ (iirc continuous with respect to the weak topologies) defined by
$$\sigma_{W}(\,^{\mathrm{t}}u(\psi), \xi)=\sigma_V(\psi,u(\xi))\; ,\; \psi\in \mathcal{V}\;,\; \xi\in W\; .$$
You can easily verify that this definition coincides with the usual one for (bounded) operators in Hilbert spaces. Dealing with unbounded operators in this setting is not so usual, but not impossible with suitable generalizations.
