Confusion about the dual/predual to the tangent plane to a Teichmüller space

Question

I apologize in advance if this question is not considered research-level.

I am reading material on Teichmüller theory and I am getting confused as to the nature of the space $Q(R)$ of all integrable, holomorphic quadratic differentials in terms of the complex structure of $\mathrm{Teich}(R)$, for a Riemann surface $R$.

More precisely, I know there is a duality between $Q(R)$ and the tangent plane to $\mathrm{Teich}(R)$ at the basepoint, but I am confused as to whether $T_0\mathrm{Teich}(R)$ is the (topological) dual of $Q(R)$ or if it is the other way around. Of course, for surfaces of finite type the question is void because then $Q(R)$ is finite dimensional. But in general $Q(R)$ is infinite dimensional and non-reflexive.

I know this is probably considered classical, but I found seemingly contradictory references on this (probably imprecisions in the use of the term "dual" due to the use of finite dimension). Can be someone please give me a definite answer, and if possible a clear reference on the topic ?

Thanks in advance.

Answer 1

For any Riemann Surface $R$, the tangent space $T_0{\rm Teich}$ is the topological dual of $Q(R)$ - whence the dual of the tangent space is the double dual of $Q(R)$. All this and more is discussed in the texts of Hubbard and Gardiner-Lakic.