Let $V$ be a vector space over a field $K$ . A linear functional (or linear form) on $V$ is a linear mapping $\phi\colon V\to K$ , where $K$ is thought of as a one-dimensional vector space over itself.

The collection of all linear functionals on $V$ can be made into a vector space by defining addition and scalar multiplication pointwise; this vector space is called the dual space of $V$ .

The term linear functional derives from the case where $V$ is a space of functions (see the entry on functionals). Some authors restrict the term to this case.