Let $H_n= \lbrace A \in M_n(\mathbb{C}), A^* = A \rbrace$ be the space of hermitian matrices, and let $iH_n= \lbrace A \in M_n(\mathbb{C}), A^* = -A \rbrace$ be the space of skew-hermitian matrices.
I've come cross the statement which says that the following expression
$$<A,B>= Im(Tr(AB)), A \in H_n, B \in iH_n, $$
defines a duality between $H_n$ and $iH_n$. (Where $Im(Tr(AB))$ is the imaginary part of the trace of the matrix $AB$).
What does the concept of duality mean in this context?
Edit: This fact was actually used to prove the following:
If $G= U(n)$ and it's Lie algebra $\mathfrak{g}$ is indentified with $iH_n$. Then using the duality above helps to prove that the co-adjoint action of $G$ on $\mathfrak{g}^*$(Which is identified with $H_n$ using that duality ) is given by conjugation: $Ad^*_{g} \xi= g \xi g^{-1}$.