The first bit, as you've seen, refers to a mapping from the Real to the Complex plane. The last condition is not referring to a vector per se (Yes, complex numbers can be seen as a vector), but the overbar (in ) refers to the complex conjugate.

Let's say for example, we have

Thus,

I leave it up to you to see if it is a vector space or not.

I hope this helps clarify things wrt notation.