linear functional LinearFunctional 2002-01-24 13:46:26 2007-01-17 05:39:35 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{scalar} \PMlinkescapeword{term} Let $V$ be a vector space over a field $K$. A \emph{linear functional} (or \emph{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 {\it linear functional} derives from the case where $V$ is a space of functions (see the entry on \PMlinkname{functionals}{Functional}). Some authors restrict the term to this case.