Sorry to ask what is almost certainly a very trivial question, but in Fulton and Harris's first course in representation theory they write down a property of root lattices which I think must be sort of wrong.
In lecture 12 they are talking about representations of $\mathfrak{sl}_3(\mathbb{C})$, and specifically show that the root lattices (lying in the plane $\mathfrak{h}^\star$) are symmetric under reflection in the line given by $\langle H_{1,2},L\rangle = 0$ where $L$ is some functional in $\mathfrak{h}^\star$. This is really confusing me (since they write a variant on this equation a few times) and it really seems to me that $H_{1,2}$ is not in $\mathfrak{h}^\star$ but in $\mathfrak{h}$, so this inner product is meaningless. If they were talking about the dual element this is the right equation, but it just seems like they've changed notation for a reason, since the dual element to this one has been written down many many times.
Is this just another example of the imprecision of this book, or am I missing something? Thanks for the help.
Let us have a finite dimensional Hilbert space $\mathfrak{h}$ (with an inner product $\langle a,b \rangle$) and its dual $\mathfrak{h}^*$.
Then $f_a(b)=\langle a,b\rangle \in \mathbb{R}$ is a linear operator.
In finite dimensional spaces those are exactly all the linear operators on $\mathfrak{h}$ wich means it is self-dual.
Thus the inner product can be defined on the dual space and with a slight abuse of notation unless specifically explained applied to elements of the the two spaces.