Definition talk:Semi-Inner Product

I have a problem with the definition section of this page:

Given the following line:

Let $V$ be a vector space over a subfield $\mathbb F$ of $\C$.

and the result Complex Numbers form Vector Space over Reals. We have that our vector space may be the complex numbers.

So the semi-inner product may be the mapping: $\innerprod \cdot \cdot: V \times V \to \mathbb C$

In other words we feed two vectors in and get a complex number out.

But then condition $(4)$ says:

$(4): \quad \innerprod x x\ge 0$

which doesn't make sense based on Complex Numbers cannot be Ordered Compatibly with Ring Structure.

Again, this stuff is way over my head. But it's an important definition and I felt like I should say something. --Jshflynn (talk) 10:07, 28 September 2012 (UTC)

Yes, sorry about that. I was trying to generalise stuff without contemplation at the time. The whole functional analysis section (of which this is about the first baby step) is due for reworking after more basic fields have been treated (I stopped mainly because there was a lack of internal references I could use; as my book is quite advanced, it is almost impossible to provide the desired standard of rigour at the moment). For now, I have added a "questionable" mark to indicate that the page may not be correct. --Lord_Farin (talk) 13:49, 28 September 2012 (UTC)

Bilinear or sesquilinear

Presently the article says the semi-inner product satisfies the property "bilinearity: ∀x,y,z∈V,∀a∈F: ⟨ax+y,z⟩=a⟨x,z⟩+⟨y,z⟩", but this formula only specifies linearity in the first argument, and then it follows from the conjugate symmetry that the semi-inner product is conjugate linear in the second argument.

In the section about spaces over real scalars, it is stated that in such spaces the conjugate symmetry becomes symmetry. I could be added that in such spaces the sesquilinearity becomes bilinearity. Cacadril (talk) 18:23, 23 February 2015 (UTC)

Okay this stuff is in advance of what I've learned formally, so my knowledge of these definitions is limited.

Suggest we add a page to $\mathsf{Pr} \infty \mathsf{fWiki}$ defining sesquilinearity in order that we have a definitional basis to hang this on. --prime mover (talk) 19:57, 23 February 2015 (UTC)

Cacadril was most correct, the page was already up, and I have edited accordingly. Thanks for noticing, it has been an omission from my side from the start. — Lord_Farin (talk) 22:21, 23 February 2015 (UTC)