As defined on Wikipedia here and there, a real structure on a complex vector space is an antilinear involution $\sigma: V\to V$ such that
- $\sigma^2=id_V$
- $\sigma(\lambda x)=\bar\lambda \sigma(x)$.
My question is : does it exist on every complex vector space ?
If $V$ has a basis $\{e_i\}_{i\in I}$, I can build this one :
$$\sigma(\sum_{i\in I}\lambda_i e_i)=\sum_{i\in I}\bar\lambda_i e_i$$
But the existence of a basis requires the axiom of choice, and I suspect that there should exist a more direct proof for the existence of a real structure.
Context My ultimate purpose is to prove the One-dimensional dominated extension theorem in the complex case.
First, I'd argue that your formula $\sigma(\sum_{i\in I}\lambda_i e_i)=\sum_{i\in I}\bar\lambda_i e_i$ is already an incredibly direct proof of the existence of such a structure.
Second, I don't think there is a 'canonical choice' for $\sigma$ that does not rely on choosing a basis. For example, if $V = \mathbb{C}$, one could consider either of $$\sigma_1(\mu) = \overline{\mu}$$ or $$\sigma_2(\mu) = -\overline{\mu}.$$ Both are real structures on $\mathbb{C}$, but they are not the same.
Essentially you have to choose in $V$ a 'real' direction and an 'imaginary' direction, but before choosing a basis there is no canonical way to do so. (The $\sigma_1$ and $\sigma_2$ essentially correspond to choosing either $1$ or $i$ as a complex basis of $\mathbb{C}$.)