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Let us call an involution on a complex linear space $X$ an arbitrary $\mathbb R$-linear map $x\in X\mapsto x^*\in X$ that satisfies the following identities: $$ x^{**}=x,\qquad (\lambda\cdot x)^*=\overline{\lambda}\cdot x^*\qquad (\lambda\in{\mathbb C},\quad x\in X). $$ I can't find a textbook on linear algebra where this notion is considered. Can anybody recommend something? I need a reference for some elementary facts like "$X$ is a complexification of the subspace of real elements" (i.e. elements satisfying the equality $x^*=x$), or "the dual space (of $\mathbb C$-linear functionals) also has a natural involution", etc.

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I think you want a book that has coordinate-free definition of a complex structure on a vector space. –  zyx Jul 6 '13 at 19:04
    
@zyx: this is strange, I can't find anything –  Sergei Akbarov Jul 6 '13 at 19:31

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