Csdrhrt

I need to know if the following reasoning is correct, if there is someone so kind to check it.

Let $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous Sobolev spaces $H^alpha$, $alphainmathbb{R}$, defined in terms of Fourier series by
$$H^alpha = left{f=sum_{kinmathbb{Z}^d} f_k, e_k, f_kinmathbb{C} Big| Vert fVert_{H^alpha}^2:=sum_k vert f_kvert^2(1+vert kvert^2)^alpha<inftyright} $$
This is a family of Hilbert spaces such that $H^alpha$ is compactly embedded in $H^beta$ whenever $alpha>beta$. Now consider the space $H^{-}:=cap_{alpha>0} H^{-alpha}$, endowed with the topology induced by the family of seminorms ${VertcdotVert_{H^{-alpha}}}_{alpha>0}$. Since the $H^{-alpha}$ are contained each one into the other, we can actually only consider a countable amount of them, say $alpha_ndownarrow 0$ and $H^-=cap_n H^{-alpha_n}$, and write an explicit metric which induces the above topology:
$$d(f,g)=sum_n 2^{-n} dfrac{Vert f-gVert_n}{1+Vert f-gVert_n}, quad VertcdotVert_n = VertcdotVert_{H^{-alpha_n}}$$
Now I claim that this space enjoys the Schur property. Let ${f_n}_n$ be a weakly convergent sequence in $H^-$, $f_n rightharpoonup f$, then it must also be weakly convergent in $H^{-alpha_n}$ for each $n$. But since $H^{-alpha_{n+1}} $ is compactly embedded into $H^{-alpha_n}$, weak convergence in the former implies strong convergence in the latter and therefore $f_nto f$ in $H^{-alpha}$ for each $alpha$, which implies strong convergence in $H^-$ as well.

I'd just like to know if this is right since when I first considered this space I was expecting it to be reflexive, but at this point it clearly can't be.