Definition:Linear Functional

Definition
Let $E$ be a vector space over $\R$ or $\C$.

Let $L$ be a linear subspace of $E$.

A mapping $f : L \to \R$ (resp. $f : L \to \C$) is called a linear functional if
 * $ f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$

holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\R$ (resp. $\C$).