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: $$ x^{**}=x,\qquad (\lambda\cdot x)^*=\overline{\lambda}\cdot x^*\qquad (\lambda\in{\mathbb C},\quad x\in X). $$ This is strange, but 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", and so on.

I posted this in math.stackexchange, but without success.

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: $$ x^{**}=x,\qquad (\lambda\cdot x)^*=\overline{\lambda}\cdot x^*\qquad (\lambda\in{\mathbb C},\quad x\in X). $$ This is strange, but 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", and so on.

I posted this in math.stackexchange, but without success.