$\mathbb{R}$-linear and $\mathbb{C}$-linear What does it mean for a map $\phi$ to be $\mathbb R$-linear or $\mathbb{C}$-linear? I have tried to look for this definition on Google but I cannot find it.
 A: For example, $\mathbb C$ is both a real vector space and a complex vector space.  The "complex conjugate" map $z \mapsto \overline{z}$ is real-linear but not complex-linear.  
explanation
Of course it is additive:
$$
\overline{z+w} = \overline{z}+\overline{w}
$$
and real scalars are OK
$$
\overline{rz} = r\overline{z}\qquad\text{if $r$ is real}
$$
But complex scalars are sometimes not OK, for example
$$
\overline{iz} = i\overline{z}
$$
is false when $z \ne 0$.
A: A linear map is a map between vector spaces (over the same field) that preserves the vector operations of scalar multiplication and vector sum. The names $\mathbb{R}$-linear and $\mathbb{C}$-linear are just a reminder of what field we are looking at.
A: In either case, we have $\phi(x + y) = \phi(x) + \phi(y)$, but the difference is under what conditions we can pull scalars out; for what scalars $k$ we are guaranteed to have $\phi(kx) = k\phi(x)$.
For a $\Bbb C$-linear map, this happens whenever $k \in \Bbb C$, and for an $\Bbb R$-linear map, this happens whenever $k \in \Bbb R$.
Source: Remmert's Theory of Complex Functions.
