Functionals (based on domain and codomain)

Question

Why is the following not a functional?

$$f(x,y)=(2x, 4y) \text{ for } V=\mathbb{R}^2$$

I know $f$ is linear, so I'm assuming there is a problem with $V$, the vector space, going to $F$, a field.

Answer 1

A functional is a linear map to $\mathbb{R}$ in the context of $\mathbb{R}^n$.

Is $(2x,4y)$ a real number?

Answer 2

If $V$ is a vector space over a field $F$, then a functinal is a linear mappin $f:V \to F$.

Your $f$ above is a mapping $f: \mathbb R^2 \to \mathbb R^2$.