I Elementary question about comparing notations of inner product

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The discussion clarifies the notations for inner products in finite vector spaces over a complex field, highlighting three interpretations: physicists' notation (a), some mathematicians' notation (b), and the bra-ket notation (c). The user questions the linearity properties of these notations, particularly how the second argument's linearity in the bra-ket notation seems to imply linearity in the first argument for the physicists' version, which leads to confusion. They acknowledge that their initial notation (a) may not be valid and seek confirmation that the bra-ket notation <v|u> aligns with the mathematicians' version (b), both representing u*v. The conversation emphasizes the importance of understanding the context of complex vector spaces in these notations. The user expresses gratitude for the patience shown in addressing their fundamental inquiries.
nomadreid
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(finite vector spaces) 3 notations: (a) physicists and others: (u,v)=v*u linear in v. (b) some mathematicians:(u,v)=u*v linear in u. (c) bra-ket: <v|u>= (u,v) from (a), so v*u, but linear in u. Seems to contradict.
First, I need to check that I have the 3 notations correct for an inner product in finite vector spaces over a complex field; v* means: given the isomorphism V to V* then:
(a) physicists and others: (u,v)=v*u ; linear in the second argument
(b) some mathematicians: (u,v)=u*v; linear in the first argument.
(c) bra-ket: <v|u>= (u,v) from (a), so v*u . <v|u> is linear in the second argument.

If these are correct, then it would seem that <v|u> being linear in the second argument (u) would imply that it would be linear in the first argument (u) for the physicist's version (a), reducing it to (b). But that is wrong. What is my confusion?

Edit: according to

https://en.wikipedia.org/wiki/Riesz...cs_notations_and_definitions_of_inner_product

my (a) does not exist. I cannot give a source for (a), as I saw this and noted it down some time ago without noting the sources. Is the solution therefore that <v|u>= (u,v) from (b)? That is, that they are both u*v ?

(Thanks for the patience with elementary questions like this one.)
 
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Order does not matter for real number inner products. However, in (c), $$<u|v> = <v|u>^*$$a conjugate complex.
 
anuttarasammyak, thank you for pointing out that I should have made precise that I was referring to complex vector spaces. I have edited the question accordingly. However, the question still stands.
 
All expressions are additive in both arguments. Whether you consider the left one or the right one as a covector is a deliberate decision, as is sesquilinearity, i.e., whether conjugating scalars in the first or second argument while pulling them out. It is similar to whether to chose (+,-,-,-) or (-,+,+,+) as Minkowski signature.

Mathematicians and physicists often use opposite conventions, but I have no idea why. It has likely historical reasons.
 

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First, I need to check that I have the 3 notations correct for an inner product in finite vector spaces over a complex field; v* means: given the isomorphism V to V* then: (a) physicists and others: (u,v)=v*u ; linear in the second argument (b) some mathematicians: (u,v)=u*v; linear in the first argument. (c) bra-ket: <v|u>= (u,v) from (a), so v*u . <v|u> is linear in the second argument. If these are correct, then it would seem that <v|u> being linear in the second...
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