I Elementary question about comparing notations of inner product

[nomadreid]

TL;DR

(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.)

 

[anuttarasammyak]

Order does not matter for real number inner products. However, in (c), $$<u|v> = <v|u>^*$$a conjugate complex.

 

[nomadreid]

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.

 

[fresh_42]

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.

 

[nomadreid]

Thanks, fresh_42.
I know that it's pure convention, but I am trying to understand the relations between the conventions.
What is clear is that
---physicists often use bra-ket <v|u>, linear in the second argument and
---mathematicians often use (u,v) , linear in the first argument,

What is not clear to me is whether
---physicists also use the notation (u,v), linear in the second argument.
If not, then all is well.
(I found my source for the latter notation: it was perhaps the worst source possible, Google's AI response to a question.)

 

[fresh_42]

Thanks, fresh_42.
I know that it's pure convention, but I am trying to understand the relations between the conventions.
What is clear is that
---physicists often use bra-ket <v|u>, linear in the second argument and
---mathematicians often use (u,v) , linear in the first argument,

I don't think that this can be said in such a rigorous way. Yes, physicists use the bra-ket notation, and mathematicians usually use the parentheses. I don't know where the conjugates appear normally. I have a right-left-weakness. I can't even tell a left-module (coset) from a right-module (coset) because I cannot remember whether left and right refer to the module or the ring (subgroup). Sesquilinearity has to be looked up in every case anyway, since you can never know for sure which author defines it how. The languages between physicists and mathematicians differ a lot. E.g., covector is a rare term in mathematics; it is a linear or dual form. Also, co- and contravariant apply only to homological algebra and functors within mathematics, not tensors. And there in the opposite direction. And what is an infinitesimal generator? A strange term for a tangent vector in the ears of a mathematician.

Conventions have to be confirmed from book to book. I usually read them from their usage and hope it is consistent throughout the book.

What is not clear to me is whether
---physicists also use the notation (u,v), linear in the second argument.
If not, then all is well.

I hope not. That would be an overstraining of notation. You can choose how to write an inner product, but you shouldn't use two different notations, and even less so code left and right with two different notations; particularly since there is no reason to use both definitions of sesquilinearity at the same time.

 

[nomadreid]

Thanks, fresh_42!