Why is inner product denoted like this?

Question

Why the inner product is denoted in every book like this: $\langle,\rangle$ instead of some name like $IP\colon V \times V \rightarrow F$? And what does the notation $\langle|\rangle$ and $\langle\cdot,\cdot\rangle$ have to do with this? Can someone explain?

Answer 1

It's just historical notational conventions from mathematics and physics, much like any piece of notation in mathematics.

Things like $\langle|\rangle$ or $\langle,\rangle$ or $\langle\cdot|\cdot\rangle$ or $\langle\cdot,\cdot\rangle$ are used to be able to denote/view/think of the inner product as a map on the product space $V\times V$. So you might write things like:

\begin{align} \langle\cdot,\cdot\rangle:V\times V&\rightarrow F\\ (v,v')&\mapsto \langle v,v'\rangle. \end{align}

To create even more confusion, sometimes "round brackets" $(\cdot,\cdot)$ are also used to denote inner products..