Linear form on arvau

Wikipedia / Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).

If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , or, when the field ;, When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).

Examples

The constant zero function, mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (that is, its range is all of ).

Mean
Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
Net present value of a net cash flow is the discount rate

Linear functionals in Rn

and each linear functional can be expressed in this form.

Trace of a square matrix

The trace is the sum of all elements on its main diagonal. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a vector space

(Definite) integration

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

Evaluation

be the evaluation functional

form a basis ( proves this last fact using Lagrange interpolation).

Non-example

having the equation of a line) is , since it is not linear. It is, however, affine-linear.

Visualization

In finite dimensions, a linear functional can be visualized in terms of its level sets, the sets of vectors which map to a given value. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes. This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as Gravitation by .

Applications

Application to quadrature

distinct points in defined above form a basis of the dual space of The integration functional is also a linear functional on This forms the foundation of the theory of numerical quadrature.

In quantum mechanics

Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are anti–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.

Distributions

In the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functionss.

Dual vectors and bilinear forms

Every non-degenerate bilinear form induces an isomorphism

(for instance, in Euclidean space is the dot product).

is said to be the dual vector.

In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem into its .

Relationship to bases

Basis of the dual space

Let the vector space , not necessarily orthogonal. Then the dual space called the dual basis

is the Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.

can be expressed as a linear combination of basis functionals, with coefficients ("components")

So each component of a linear functional can be extracted by applying the functional to the corresponding basis vector.

The dual basis and inner product

When the space carries an inner product, then it is possible to write explicitly a formula for the dual basis of a given basis. Let In three dimensions ( where ε is the Levi-Civita symbol the inner product (or dot product) on .

is the Hodge star operator.

Over a ring

Modules over a ring are generalizations of vector spaces, which removes the restriction that coefficients belong to a field. Given a module over a ring , a linear form on is a linear map from to , where the latter is considered as a module over itself. The space of linear forms is always denoted , whether is a field or not. It is a right module if is a left module.

The existence of "enough" linear forms on a module is equivalent to projectivity.

Change of field

gives rise to a real vector space called the endowed with a complex structure; that is, there exists a real vector subspace-vector spaces.

Real versus complex linear functionals

), where the image denotes the space's algebraic dual space is an \R-linear operator (meaning that it is additive and \R-linear -linear functional as well.

Real and imaginary parts

then denote its real part and its imaginary part

defines a bijective

This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray), and can be generalized to arbitrary finite extensions of a field in the natural way. It has many important consequences, some of which will now be described.

Properties and relationships

is a topological vector space are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word "bounded where the prime denotes the space's continuous dual space.

of unit length) then is a normed space is the closed unit ball then the supremums above are the operator norms so that This conclusion extends to the analogous statement for polars of balanced sets in general topological vector spaces.

is a complex Hilbert space with a (complex) inner product that is antilinear Applying the Riesz representation theorem which is the same conclusion that was reached above.

In infinite dimensions

Below, all vector spaces are over either the real numbers or the complex numbers

is a topological vector space, the space of continuous linear functionals — the is a Banach space, then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the . In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous dual is a proper subspace of the algebraic dual.

A linear functional on a (not necessarily locally convex) topological vector space is continuous if and only if there exists a continuous seminorm on

Characterizing closed subspaces

Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed, and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.

Hyperplanes and maximal subspaces

is called maximal that is not identically ). An affine hyperplane

Relationships between multiple linear functionals

Any two linear functionals with the same kernel are proportional (i.e. scalar multiples of each other). This fact can be generalized to the following theorem.

If is a non-trivial linear functional on with kernel and is a balanced subset of

Hahn–Banach theorem

Any (algebraic) linear functional on a vector subspace However, this extension cannot always be done while keeping the linear functional continuous. The Hahn–Banach family of theorems gives conditions under which this extension can be done. For example,

Equicontinuity of families of linear functionals

Let be a topological vector space (TVS) with continuous dual space

For any subset the following are equivalent:

is equicontinuous;
is contained in the polar in ;
the (pre)polar of in ;

If then the following sets are also equicontinuous: the weak-* closure, the balanced hull, the convex hull, and the convex balanced hull. Moreover, Alaoglu's theorem is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).