Definition:Linear Functional

Definition

Let $E$ be a vector space over $\Bbb F \in \left\{{\R, \C}\right\}$.

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

A mapping $f : L \to \Bbb F$ is called a linear functional iff

$ f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$

holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\Bbb F$.

Notation

It is customary in functional analysis to omit the parentheses around the argument of linear functionals whenever this can be done unambiguously.

For example, one commonly writes $f(\alpha x+y) = \alpha fx + fy$ to denote the linearity of $f$ as described above.

This is done to avoid tedious writing of parentheses, and make formulae look more appealing.

It is therefore important to strongly keep in mind which letters denote scalars, functionals and elements of the vector space, so as to avoid confusion.