Linear functional

Linear functional

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In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, a linear functional or linear form (also called a one-form
One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...

or covector) is a linear map from a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 to its field of scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

s.  In Rn
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

, or the matrix product with the row vector on the left and the column vector on the right.  In general, if V is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 k, then a linear functional ƒ is a function from V to k, which is linear: for all for all

The set of all linear functionals from V to k, Homk(V,k), is itself a vector space over k.  This space is called the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space.  It is often written V* or when the field k is understood.

Continuous linear functionals


If V is a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

, the space of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 linear functionals — the continuous dual — is often simply called the dual space.  If V is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

, then so is its (continuous) dual.  To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual.  In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions.

Linear functionals in Rn


Suppose that vectors in the real coordinate space Rn are represented as column vectors


Then any linear functional can be written in these coordinates as a sum of the form:


This is just the matrix product of the row vector [a1 ... an] and the column vector :

Integration


Linear functionals first appeared in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, the study of vector spaces of functions
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

.  A typical example of a linear functional is integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

: the linear transformation defined by the Riemann integral
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...




is a linear functional from the vector space C[a,b] of continuous functions on the interval [ab] to the real numbers.  The linearity of I(ƒ) follows from the standard facts about the integral:

Evaluation


Let Pn denote the vector space of real-valued polynomials of degree ≤n defined on an interval [a,b].  If c ∈ [ab], then let evc : Pn → R be the evaluation functional:
The mapping ƒ → ƒ(c) is linear since

If x0, ..., xn are n+1 distinct points in [a,b], then the evaluation functionals evxi, i=0,1,...,n form a basis of the dual space of Pn.  ( proves this last fact using Lagrange interpolation.)

Application to quadrature


The integration functional I defined above defines a linear functional on the subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

 Pn of polynomials of degree ≤ n.  If x0, …, xn are n+1 distinct points in [ab], then there are coefficients a0, …, an for which


for all ƒ ∈ Pn.  This forms the foundation of the theory of numerical quadrature.

This follows from the fact that the linear functionals evxi : ƒ → ƒ(xi) defined above form a basis of the dual space of Pn .

Linear functionals in quantum mechanics


Linear functionals are particularly important in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

.  Quantum mechanical systems are represented by Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s, 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
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

.

Distributions


In the theory of generalized function
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...

s, certain kinds of generalized functions called distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

 can be realized as linear functionals on spaces of test functions.

Properties

  • Any linear functional is either trivial (equal to 0 everywhere) or surjective onto the scalar field.  Indeed, this follows since the image of a vector subspace
    Linear subspace
    The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

     under a linear transformation is a subspace, so is the image of V under L.  But the only subspaces (i.e., k-subspaces) of k are {0} and k itself.

  • A linear functional is continuous if and only if its kernel is closed .

  • Linear functionals with the same kernel are proportional.

  • The absolute value of any linear functional is a seminorm on its vector space.

Dual vectors and bilinear forms



Every non-degenerate bilinear form on a finite-dimensional vector space V gives rise to an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 from V to V*. Specifically, denoting the bilinear form on V by ( , ) (for instance in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 (v,w) = vw is the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

 of v and w), then there is a natural isomorphism given by


The inverse isomorphism is given by where ƒ* is the unique element of V for which for all w ∈ V


The above defined vector v* ∈ V* is said to be the dual vector of v ∈ V.

In an infinite dimensional Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

, analogous results hold by the Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

.  There is a mapping V → V* into the continuous dual space V*.  However, this mapping is antilinear rather than linear.

Visualizing linear functionals


In finite dimensions, a linear function can be visualized in terms of its level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....

s.  In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

s.  This method of visualizing linear functionals is sometimes introduced in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 texts, such as .

Basis of the dual space in finite dimensions


Let the vector space V have a basis , not necessarily orthogonal.  Then the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 V* has a basis called the dual basis defined by the special property that
Or, more succinctly,
where δ is the Kronecker delta.  Here the superscripts of the basis functionals are not exponents but are instead contravariant
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

 indices.

A linear functional belonging to the dual space can be expressed as a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of basis functionals, with coefficients ("components") ui ,
Then, applying the functional to a basis vector ej yields
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals.  Then
that is
This last equation shows that an individual component of a linear functional can be extracted by applying the functional to a corresponding basis vector.

The dual basis and inner product


When the space V carries an inner product, then it is possible to write explicitly a formula for the dual basis of a given basis.  Let V have (not necessarily orthogonal) basis .  In three dimensions (n = 3), the dual basis can be written explicitly
for i=1,2,3, where is the Levi-Civita symbol
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

 and the inner product (or dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

) on V.

In higher dimensions, this generalizes as follows
where is the Hodge star operator.