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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.

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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-Alvarez Feb 24 '11 at 2:02
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 2:56
up vote 9 down vote accepted

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

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.

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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).

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