\begin{definition}[Definition:Linear Functional]
Let $E$ be a vector space over a field $\Bbb F$.
Let $D$ be a linear subspace of $E$.
A mapping $f : D \to \Bbb F$ is called a '''linear functional''' {{iff}}:
:$\map f {\alpha x + \beta y} = \alpha \map f x + \beta \map f y$
holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\Bbb F$.
\end{definition}