I'm looking at a definition for linear functionals on a vector space. It says it's a scalar valued function and then states the linearity condition. I've just been looking at some exercises in a book and I was narrowing down which ones were functionals.
question: when it says 'scalar valued' does that mean that if $V$ is a vector space over a field $\cal F$ then a linear functional should be a map $y:V\rightarrow \cal F $. Or can it map to any field?
I'd assume not because you'd have to make sense of $\alpha\cdot y (x)$ where $\alpha \in \cal F$ and $x \in V$ otherwise.
In which case the map $y$ from $\Bbb C$ as a real vector space to $\Bbb C$ via complex conjugation would be ruled out.