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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?

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    $\begingroup$ I don't know the historical reason for the notation, but there are many instances of binary functions where an infix notation is more convenient than a prefix notation. Even simpler examples include addition of real numbers, which is really a function $\mathrm{add} : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, but writing $x+y$ instead of $\mathrm{add}(x,y)$ (or even $a(x,y)$ or whatever) clarifies matters considerably e.g. the associativity axiom: $x+(y+z)=(x+y)+z$ versus $a(x,a(y,z))=a(a(x,y),z)$. $\endgroup$ – Clive Newstead Sep 11 '17 at 1:23
  • $\begingroup$ I don't know the answer either, but I like that^ response. $\endgroup$ – Randall Sep 11 '17 at 1:25
  • $\begingroup$ Also, well. (1) Inner product is sometimes written in the usual functional way, when it's to emphasize it is indeed a function (e.g. in communication complexity); (2) Is your middle question referring to the "bra-ket" notation of inner product in quantum physics? $\endgroup$ – Clement C. Sep 11 '17 at 1:34
  • $\begingroup$ I suggest the tags math-notation and math-history if that is what you are interested in $\endgroup$ – mdave16 Sep 11 '17 at 1:41
  • $\begingroup$ @ClementC. Yes, I am too referring to the bra-ket notation but I have no idea how it is used here and if there is any difference. Disclaimer: I am not familiar with quantum mechanics and bra-ket notation. $\endgroup$ – LearningMath Sep 11 '17 at 11:11
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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..

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    $\begingroup$ Yes the dots just indicate where the arguments of the inner product go in the notation, so it is kind of like writing $f(\cdot,\cdot)$ for a function of two variables. $\endgroup$ – g.s Sep 11 '17 at 20:36
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    $\begingroup$ A finite-dimensional vector space $V$ is linearly isomorphic to its dual space, so every linear functional $f:V\rightarrow\mathbb{R}$ given by $v\mapsto f(v)$ on $V$ can be identified with a unique element $v^*$ of $V$ via the inner product, in which case we have $f(v)=\langle v^*,v\rangle=\langle v^*|v\rangle$. The bra-ket notation makes this identification easy to write down: we simply write $f=\langle v^*|$ and thus think of $\langle v^*|$ as the linear functional. $\endgroup$ – g.s Sep 11 '17 at 20:41
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    $\begingroup$ It's also common to think of $v^*$ itself as the linear functional and write $v^*(v)$ for $\langle v^*,v\rangle$ or $\langle v^*|v\rangle$, etc, whatever notation you're using. $\endgroup$ – g.s Sep 11 '17 at 20:45
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    $\begingroup$ $v^*$ is viewed or thought of as a linear functional since the map $v\mapsto\langle v^*,v\rangle$ is the actual linear functional, and every linear functional on $V$ can be uniquely identified with an element of $V$ in this way. $\endgroup$ – g.s Sep 11 '17 at 20:55
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    $\begingroup$ I just did above (the association is made via the inner product), and it is also explained in most linear algebra books: You prove that the inner product space $V$ and its dual $V^*$ have the same dimension and the map $V\mapsto V^*$ given by $v^*\mapsto \langle v^*,\cdot\rangle=\langle v^*|$ is a linear space isomorphism. Also consider the Riesz representation theorem, which is the analogous statement for infinite dimensional (Hilbert) spaces. $\endgroup$ – g.s Sep 11 '17 at 21:56

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