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Let $\mathcal{X}$ be a vector space on some field $\mathbb{K}$.

Most often, when we want to give it a dot product it is in the form of a bilinear definite positive, symmetric application $\mathcal{X}\times\mathcal{X}\mapsto\mathbb{R}$.

But if $\mathbb{K}$ is $\mathbb{C}$, the definition changes to hermitian, sesquilinear and the positivity should be changed from $\langle x|x \rangle \geq0$ to $\langle x|x \rangle\in\mathbb{R}^+$ (which I never saw expressed like this, the comparison to $0$ was always implicitly considered valid).

Is there a more generic definition that encompass all possible $\mathbb{K}$, possibly with adding somme requirements to $\mathbb{K}$ (that seems necessary in order to define positivity)? Besides, must the output be in the same field that the one upon which is defined $\mathcal{X}$?

IMHO, the right idea is "inner product" but the definitions I can find do not answers the questions above and seem even wrong with respect to the type of symmetry.

Note: closest questions I found are:

But their answers are not complete enough (though the third one introduced the idea of using an involution):

  • how does positivity generalized
  • how to derive a norm, have Pythagore and Cauchy-Schwartz
  • is the product $\mathcal{X}\times\mathcal{X}\mapsto\mathbb{K}$ or can we switch to $\mathbb{K'}$
Oersted
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  • This was linked in one of your links. I think they make some important observations, and I agree with the idea that an "inner product" ought to be an anisotropic quadratic form. Obviously that doesn't work directly for $\mathbb C$, so I would suggest maybe that an "inner product" on a $K$-vector space $V$ consists of an involution $$ on $K$ and a $$-sesquilinear form on $V$ which is anisotropic when considered as a symmetric $L$-bilinear form, where $L$ is the fixed field of $*$. – Nicholas Todoroff Dec 12 '24 at 16:05
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    Please specify exactly what is missing in the links you have. https://math.stackexchange.com/questions/3625701/what-properties-must-a-field-k-satisfy-in-order-to-define-an-inner-product-on seems to be a 100% duplicate for example. "Answers are not complete enough" is not specific and won't help potential answerers to address your problem. – Martin Brandenburg Dec 12 '24 at 17:29
  • Does this answer your question? Option 1 is what I suggested, and they give lots of references. – Nicholas Todoroff Dec 12 '24 at 19:25
  • @NicholasTodoroff this is interesting but without full access to the document I cannot check some claims. For instance, the fact that Cauchy-Schwarz is till valid, seems to rely on additional axioms. (looking to a few proofs, that relies on polynomial expressions, I think that it restricts the possibilities on the output space of the inner product). Also what is <lx|y> (l<x|y>? if (lx)* is lx for the automorphism). How to derive a norm? Sorry, not much place to expose my concerns. – Oersted Dec 13 '24 at 16:23
  • @Oersted Yes, you're right, that's quite a glaring oversight by the author of the answer. I had a look at the book, $K$ is assumed to be a "complete non-Archimedean valued field with nontrivial valuation". This means $K$ is equipped with a nontrivial valuation that induces an ultrametric which is complete. I have no idea what you mean by "what is $\langle lx|y\rangle$?" – Nicholas Todoroff Dec 13 '24 at 17:26
  • @NicholasTodoroff not easy to express these subjects in comments. ⟨lx|y⟩ was about how linearity or sesquilinearity can be expressed in a generalised way. For instance an involution f(lx) can give g(l)f(x) where g(l) is also an involution on K? – Oersted Dec 16 '24 at 09:27
  • Sesquilinearity is (1) linear in one argument (let's say second)$$\langle x,ay+by'\rangle=a\langle x,y\rangle+b\langle x,y'\rangle;$$(2) conjugate symmetry$$\langle x,y\rangle=\langle y,x\rangle^.$$These together imply conjugate linearity in the first argument,$$\langle ax+bx',y\rangle=\langle y,ax+bx'\rangle^=a^\langle y,x\rangle^+b^\langle y,x'\rangle^=a^\langle x,y\rangle+b^\langle x',y\rangle.$$I was confused because your question came out of nowhere. – Nicholas Todoroff Dec 16 '24 at 15:39

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