In the J. Tate's paper "Relations Between K2 and Galois Cohomology" Lemma 5.2, it says that given $\alpha_1,\cdots,\alpha_r\in $Br$_lF$, i.e. $\alpha_i$ is killed by $l$, where $l$ is prime and $l\neq$ char $F$ and $F$ is a global field. If there exists for each place $v$ of Fan element $x_v\in F_v^{\times}$ such that $(a_i,x_v)=(\alpha_i)_v$ for each i, then there exists an element $x\in F^{\times}$ such that $(a_i,x)=\alpha_i$ for each i. In the proof, he let $J$ is the idele group of $F$. And he says that $J/J^{l}$ is its own Pontryagin dual with respect to the pairing $$<\xi,\eta>=\sum\limits_v\mbox{inv}_v(\xi_v,\eta_v)_v,$$ where $(\xi_v,\eta_v)_v$ is the cyclic algebra over $F_v$. That is where I'm confused. I can't see why it is its own dual.
I guess that we need to prove $J/J^l\cong \widehat{J/J^l}$ by showing $$\eta\mapsto(\xi\mapsto<\xi,\eta>)$$ is isomorphic. I don't know how to use the interrelationships of global and local class field theory and Kummer theory.
Any hint?
Thanks.