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In my measure theory and integration course we learned about line integrals and surface integrals. My lecturer uses the notation $(u | v)_{\mathbb{R}^n}$ to denote the (Euclidean) dot product of two vectors $u, v\in \mathbb{R}^n$, which is a notation that I have never seen before except for the books of Daniel Stroock on integration (books that my lecturer mentioned and recommended to us-otherwise I would have never heard of these books since I don't think they are that popular, but I may be wrong since I am a novice when it comes to measure, integration and related aspects). This means that if we, say, integrate some function $F$ on some space curve $\sigma$, then the notation used would be $$\int_{\sigma} (F|dl)_{\mathbb{R}^3}$$ due tot the way we denote the dot product.
Needless to say that in any online resource (and any books I have) this is not the way to denote things.
So, is this notation popular or is it some weird one used by only a handful of people? My hunch is that it is common in certain areas of mathematics (I suspect PDEs/functional analysis because this is my lecturer's research field, but I haven't taken any course on either of those subjects, so I really can't say if my hunch is correct) and certainly not at the basic level (i.e. a regular multivariable calculus class let's say), this being the reason that it seemed weird to me.

RobPratt
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MathIsCool
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    It is not especially common, but it is not weird. – copper.hat Dec 20 '21 at 22:00
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    Indeed not common. https://mathoverflow.net/questions/366070/what-are-the-benefits-of-writing-vector-inner-products-as-langle-u-v-rangle/366118 –  Dec 20 '21 at 22:03
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    I would say the notation comes more from physics, quite related to the bra- and -ket. – Ted Shifrin Dec 20 '21 at 22:10
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    Actually, I have already seen this notation a couple of times, so i don't find it weird at all. In fact, in my first higher dimensional analysis classes we always denoted the line integrals like that... – G. Blaickner Dec 20 '21 at 22:11
  • @copper.hat thanks, I realize now that weird may have been a tad too strong. I think that I wanted that weird to mean uncommon. – MathIsCool Dec 20 '21 at 22:20
  • @TedShifrin interesting, thank you! I hadn't heard about this since I am not into physics, but I see that this bra-ket notation seems to be a good reason to go for this notation for inner products. – MathIsCool Dec 20 '21 at 22:22
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    @G.Blaickner yeah, I do realize that I should have said uncommon instead of weird. I am glad that it is not such a strange notation after all since other people also use it :). – MathIsCool Dec 20 '21 at 22:23
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    It is not uncommon, but best avoided, in my opinion. I believe the only reason it is used is to avoid having to write \langle and \rangle over and over again, but with the new command declaration utilities of Latex this really is a non issue. – K.defaoite Dec 20 '21 at 23:22
  • To expand on K.defaoite's comment, one Should code LaTeX semantically, introducing a macro such as \IP{u}{v}. Doing so saves keystrokes, is less error-prone, and allows one's code to be adjusted to a house style by changing one instance in the preamble, without editing however many dozens of instances in the document body. – Andrew D. Hwang Dec 21 '21 at 03:06

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