Reading an old paper of Weil's (translation: On certain groups of unitary operators), I'm confused about what should be a rather basic point.
Let $G$ be a locally compact abelian group. Now in addition he assumes that $G$ is isomorphic to its dual group of characters. For $x$ in $G$, let $x^*\in G$ be a character. He says that one denotes $\langle x,x^*\rangle$ the value $x^*(x)$. [I think this bracketed expression is referred to as a pairing more generally, but I'm unsure and would appreciate it if someone could explain this.] He then says one can identify $G$ with $(G^*)^*$ (which I think is just Pontryagin Duality), and says that therefore $$\langle x,x^*\rangle=\langle x^*,x\rangle.$$
Meaning that $x(x^*)=x^*(x)$ for all $x,x^*$? I don't understand how this is obvious. Does this only happen for self-dual groups? Weil seems to suggest that this follows just from $G=(G^*)^*$.