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I am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in general).

In particular, understanding of the basic theory is necessary and useful for a more intuitive approach towards functional analysis.

Can you give me a tip? For example, Serge Lang's classical book does not explicitly work this part out. I am aware of a few pages in Arnold's book on ODE, but there should be something more comprehensive and neat somewhere out there.

Hosein Rahnama
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shuhalo
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    What exactly do you want to know? I doubt there is something as impressively sounding as a "good comprehensive resource" on such a subject... I mean: there is not that much to be said! – Mariano Suárez-Álvarez Feb 24 '11 at 02:02
  • If I remember correctly, Steven Roman's Advanced Linear Algebra (Springer-Verlag) discusses the complexification of a real space. There really isn't that much to be said: given fields $F\subseteq K$, a $K$-vector space is an $F$-vector space via the forgetful functor (and a basis for $K$ over $F$ gives you all the information you need to describe $V$ as an $F$-vector space if you know $V$ as a $K$-vector space; and $V\otimes_F K$ gives you the $K$-ification of an $F$-vector space $V$; again, knowing a basis for $K$ over $F$ gives you pretty much all you need. – Arturo Magidin Feb 24 '11 at 02:56
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    @MarianoSuárez-Álvarez FYI: I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification, even though I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier. See my recent questions if you're interested. I believe there is that much to be said. – BCLC Jan 30 '20 at 08:47
  • @ArturoMagidin FYI: I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification, even though I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier. See my recent questions if you're interested. I believe there is that much to be said. – BCLC Jan 30 '20 at 08:48
  • @JohnSmithKyon: it is bad form to reply 9 years after the fact, more so when all you seem to be doing is trying to call attention to your questions, by vaguely referencing them (and complaining that your study techniques don't seem to be helping you). Worse when you repeat the same post verbatim twice, just so you can poke multiple people with the unnecessary and uninformative comments. – Arturo Magidin Jan 30 '20 at 16:32

2 Answers2

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As others have said, one can find (often brief) treatments of these issues in many places. However, an unusually thorough and insightful account is given in this exposition of Keith Conrad.

Note: "unusually" means "unusually for treatments of complexification", not "unusually for Keith Conrad". In fact I give his entire page of expository writing -- namely

http://www.math.uconn.edu/~kconrad/blurbs/

my highest recommendation: it is a treasure trove of intermediate level math. And when I say "intermediate level", I don't mean that grown up mathematicians can't profit from reading them. We sure can -- I have learned a lot, and I have not hesitated to flatter them in the most sincere way.

Pete L. Clark
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  • I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification, even though I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier. May you please help me with some of my questions on these, Pete L. Clark? I have posted at least 12 already. – BCLC Jan 30 '20 at 13:08
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Please find §12 "Complexification and Decomplexification" in book: "LINEAR ALGEBRA AND GEOMETRY" by Kostrikin & Manin (1989), pages 75-81.

There you will find an excellent answer to your question (according to my point of view).

Oleg
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    In particular, an extremely nice point that’s missing is Conrad is that $\mathbf C\otimes V_{\mathbf R}\cong V\otimes\bar V$ naturally (§12.13). That’s where the $dz$ and $d\bar z$, $\partial/\partial z$ and $\partial/\partial\bar z$ in complex analysis and complex differential geometry come from. (In fact, Conrad does not define $\bar V$ at all.) – Alex Shpilkin Mar 15 '18 at 05:43
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    @Alex: Do you mean to rotate your $\otimes$ by $\pi/2$? – Pete L. Clark Jan 11 '19 at 22:39
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    @Pete: Oh. Yes, I did (by $\pi/4$, though =) ). That should’ve been $\mathbf C\otimes V_{\mathbf R}\cong V\oplus\bar V$, that is, the complexification of realification of a space is (canonically!) isomorphic to the direct sum of the space itself and its complex conjugate. – Alex Shpilkin Jan 12 '19 at 00:29
  • Alex: Right, $\pi/4$!! – Pete L. Clark Jan 12 '19 at 21:20
  • I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification, even though I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier. May you please help me with some of my questions on these, Oleg? I have posted at least 12 already. – BCLC Jan 30 '20 at 13:09