In Atiyah's paper "K-theory and reality", p. 370, there is an example of a "real" algebraic space. Given the complex projective space $X=P(C^n)$, one considers the standard line-bundle $H$ over $X$ through the following exact sequence : $E \rightarrow X \times C^n \rightarrow H \rightarrow 0$, where $E \subset X \times C^n$ is defined as the set of pairs $(z,u) \in X \times C^n$ such that: $\sum_i u_i z_i=0$.
Then it is written: "since this equation has real coefficients, $E$ is a real bundle".
My question: I do not see what is meant here since the coefficients $u$ belong to $C^n$. Could someone explain me in what sense this equation has "real" coefficients?
(As a matter of fact the paper is hard to follow due to the use of the same word "real" for different meanings. The later use of "Real" vs "real" was a nice idea...)