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I have added some textbook references
and some historical references,
such as to von Neumann’s original definition of Hermitian forms (1930, 1932)
Von Neumann’s is the modern definition – was he the first to state it this way?
I gather from
that the concept is named after
but it is not easy to recognize a Hermitian form in there (admittedly, I have only scanned over the article, so far)
Similarly for
which may be the earliest coinage of the term “Hermitian form” (?)
I have now written out more text in the section on reagarding Hermitian/unitary complex modules as complex Euclidean/orthogonal “Real complex modules” internal to the category of real vector spaces with involutions.
Also adjusted some of the terminology, for better flow and thus renamed the sub-section:
added pointer to:
I see now that the embedding of fin-dim Hermitian spaces into self-dual $\mathbb{Z}/2 \curvearrowright \mathbb{C}$-modules which I described (here) is essentially what is known in the context of Hermitian K-theory as the hyperbolic functor
Or rather, what I consider here is the hyperbolic functor equipped with $\mathbb{Z}/2 \times \mathbb{Z}/2$-equivariance. I see now that this is known as establishing an equivalence between KR-theory and topological Hermitian K-theory.
The two references for this equivalences that I am (now) aware of (here) are:
and
both of which are rather terse, each in its own way.
So I am not surprised that my construction here is known, on the contrary, I was surprised that I couldn’t find something that elementary discussed in the literature. And with the above two references it still takes some squinting to see what’s going on.
added (here) statement and proof of a more pronounced way of phrasing the expression of Hermitian forms as inner products internal to Real vector bundles:
Passing a real inner product space $(\mathscr{V}, g)$ equipped with an isometric complex structure $J$ through the equivalence of real vector spaces with Real vector bundles over the point yields the corresponding Hermitian form $\langle - \vert - \rangle \,\equiv\, g(-,-) + \mathrm{i} g\big(J(-), v\big)$, as shown here.
added pointer to:
added pointer to:
We now have a (very) brief note on the story I was getting at above, at Quantum and Reality.
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