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The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where did this notation come from.

John P. D'Angelo's book, simply says "See [GR, p.2] for a discussion of the etymology of this standard notation." where [GR] is Grauert and Remmert's "Coherent analytic sheaves". However I have no access to that book.

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4  
You mean O, not $\Omega$, right? And "holomorphic" functions, not homomorphic ones. Anyway, back in the 19th century Dedekind used the fraktur letter O (well, more often o I think) to denote rings of algebraic integers. It came from his term Ordnung for what later became rings, more or less. Thus in complex analysis the fundamental ring of holomorphic functions became denoted with O too, but they like to think it also honors the work of Oka. :) – KConrad Mar 25 '12 at 4:41
    
Thanks, fixed. I hope I could mark this as the answer. Is this also why the structural sheaf of a manifold also shares this notation? – ssquidd Mar 25 '12 at 6:42
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I've heard that it comes from the Italian "(Funzione) Olomorfa" but have never seen it written anywhere. – Georges Elencwajg Mar 25 '12 at 10:23
    
I had heard that the O used to denoted the structure sheaf of a complex manifold, i.e. the sheaf of holomorphic functions, is because of the word "holos" (as in holo- morphic) in Greek begins with the letter $\omicron$. Wolfram Math and Wikipedia confirm this: holos = $\omicron \lambda \omicron \sigma$ (The sigma here is incorrect. I don't think Latex does the other lower case form of sigma) – Andrew McHugh Apr 19 at 15:40

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