Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$.
Let $f: H^1(G_K,E[2])\to \bigoplus_v H^1(G_{K_v},E[2])$ be a natural map.
For $M$: profinite abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bbb{Q}/\Bbb{Z})$ be the Pontryagin dual of $M$. For group hom $f$ between profinite groups, we denote $\widehat{f}$ by its dual hom.
In $1.2$ of Coates Sujata's 'Galois cohomology of elliptic curves', they states
$\text{Im}(f)=\text{ker}(\widehat{f})$
(They put $M=E[2]$ and $W_v(M)$ in the book as the image of Kummer map). Why does this hold ? In general situation, such a = does not hold (cf. Does $\text{Im}(f)\cong \text{Ker}(f^*)$ hold? Pontryagin dual).
However, in their paper, they write about this completeness as if it were a matter of course. How do they prove it?