This article deals with linear transformations from a vector space to its field of scalars. These transformations may be functionals in the traditional sense of functions of functions, but this is not necessarily true.

In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. Specifically, if V is a vector space over a field k, then a linear functional is a linear function from V to k.

The set of all linear functionals from V to k, Hom_K(V,k), is itself a k-vector space. This space is called the dual space of V. If V is a topological vector space, the space of continuous linear functionals — the continuous dual is often simply called the dual space. If V is a Banach space then so is its continuous dual.

Any linear functional is either trivial (equal to 0 everywhere) or surjective onto the scalar field. (Because the image of a vector subspace under a linear transformation is a subspace, so is the image of V under L. But the only subspaces of k are {0} and k proper).

Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral

$f\; \backslash mapsto\; \backslash int\_a^b\; f(x)\backslash ,\; dx$

is a linear functional from the space of continuous functions on the interval ^* to the real numbers.

Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra-ket notation.

A generalized function is an example of a linear functional.

The reason for the use of the term "functional" instead of the traditional term "function" is to avoid potential confusion when a vector space is a space of functions, which is often the case. Hence, linear functionals are often, in practice, functionals in the traditional sense of functions of functions.

See also

• Discontinuous linear map
• Riesz representation theorem
• Positive linear functional
• Bilinear form

Linearform | Forme linéaire | Funzionale lineare | Funkcjonał liniowy