interpretation of angle bracket $\langle \circ ; \circ \rangle$ notation appeared in linear algebra

Question

I'm confused around the angle bracket $\langle \circ ; \circ \rangle$ notation appeared in my book, where no definition of that is given. So, it is really nice if someone tell me how to interpret this. The context that the notation is used is as follows:

Definition 2.28 (Dual of a linear map)
If $A: U \to V$ is a linear map between finite-dimensional vector space $U$ and $V$, then the linear map $A^*: V^* \to U^*$ defined by

$\langle A^*(\alpha) ; u \rangle = \langle \alpha; A(u)\rangle, \alpha \in V^*, u \in U$

is a dual of $A$.

The book name is Bullo, Francesco, and Andrew D. Lewis. "Geometric Control of Mechanical Systems." (2005).

Answer 1

The notation $\langle A^*(\alpha) ; u \rangle$ is another notation for $A^*(\alpha)(u)$.

It is a notation to highlight duality.