For example in Atiyah's $KR$-theory there is the notion of a Real vector bundle in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^*$-algebra and there are probably a lot more objects with sensible definitions of Real.

However Atiyah did not use a capitalized Real in his paper "K-theory and reality", so my question is:

Whose ideas was it to write Real instead of real?

I would also be interested in the general history of this notion, but that's probably too much to ask for (judging from remarks in the above mentioned paper I suspect it starts somewhere in algebraic geometry – of which I hardly know anything.)

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    $\begingroup$ Is there any reason to think that it was not Atiyah in the article mentioned? $\endgroup$ – Mikhail Katz Apr 16 '18 at 9:59
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    $\begingroup$ I had never heard this terminology before, but it seems like an atrocious idea for a number of reasons to distinguish "real" and "Real" (how are you supposed to make the distinction orally? how are you supposed to start a sentence with "Real vector bundles"? and — of course this problem might have been unforeseeable when the term was invented — how are you supposed to search for this online? or simply, look it up in an encyclopedia?). $\endgroup$ – Gro-Tsen Apr 16 '18 at 13:35
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    $\begingroup$ Surely it would be better to write "complexified vector bundle" rather than "Real vector bundle". $\endgroup$ – Ben McKay Apr 16 '18 at 13:49
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    $\begingroup$ @Gro-Tsen, there is also the distinction in motivic integration between 'function' and 'Function'. (I agree about its desireability!) $\endgroup$ – LSpice Apr 16 '18 at 13:54
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    $\begingroup$ @Gro-Tsen When I'm discussing KR-theory with my advisor (in German) we usually say "real" instead of the German word "reell" to make the distinction … that's bearable, though a poor choice still $\endgroup$ – JBantje Apr 16 '18 at 14:57

Apparently, Atiyah himself thinks he invented the notation:

(quote from his collected works)

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    $\begingroup$ So, er, "5. The Real case" or "5. The real case"..? $\endgroup$ – David Richerby Apr 16 '18 at 16:29

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