linear form, on a vector space over a field
A mapping such that
for all , . The concept of a linear functional, as an important special case of the concept of a linear operator, is one of the main concepts in linear algebra and plays a significant role in analysis.
On the set of linear functionals on the operations of addition and multiplication by a scalar are defined according to the formulas
They specify in a vector space structure over .
The kernel of a linear functional is the subspace . If (that is, ), then is a hyperplane in . Linear functionals with the same kernel are proportional.
If is a basis of , then for
The correspondence is an isomorphism of onto . Corollary: is isomorphic to if and only if it is finite dimensional. On transition to a new basis in the elements are transformed by the same formulas as the basis vectors.
The operator defined by is injective. It is an isomorphism if and only if is finite dimensional. This isomorphism, in contrast to the isomorphism between and , is natural, i.e. functorial (cf. Functorial morphism).
A linear functional on a locally convex space, in particular on a normed space, is an important object of study in functional analysis. Every continuous (as a mapping on topological spaces) linear functional on a locally convex space is bounded (cf. Bounded operator), that is,
for all bounded . If is a normed space, the converse is also true; both properties are then equivalent to the finiteness of the number
The continuous linear functionals on a locally convex space form a subspace of , which is said to be the dual of . In one considers different topologies, including the weak and strong topologies, which correspond, respectively, to pointwise and uniform convergence on bounded sets. If is a normed space, then is a Banach space with respect to the norm and the corresponding topology coincides with the strong topology. The unit ball , considered in the weak topology, is compact.
The Hahn–Banach theorem has important applications in analysis; one formulation of it is as follows: If is a pre-norm on a vector space and if is a linear functional defined on a subspace of such that for all , then can be extended to the whole of , preserving linearity and the given bound. Corollary: Any continuous linear functional defined on a subspace of a locally convex space can be extended to a continuous linear functional on , and if is a normed space, then the norm is preserved. Hence, for every , , there is an with .
Let be a normed space and suppose that , and then , are taken with the corresponding norms. Then the operator
is an isometric imbedding. If under this imbedding coincides with , then , which is necessarily complete, is said to be reflexive (cf. Reflexive space). For example, and , , are reflexive if and only if . There is a similar concept of reflexivity for general locally convex spaces.
For many locally convex spaces, all linear functionals have been described. For example, the adjoint of a Hilbert space is . The adjoint of is .
|||N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)|
|||A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)|
|[a1]||A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)|
Linear functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_functional&oldid=14314