Linear form


In mathematics, a linear form is a linear map from a vector space to its field of scalars.
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 is understood, ; other notations are also used, such as, or When vectors are represented by column vectors, then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products.

Examples

The constant zero function, mapping every vector to zero, is trivially a linear functional. Every other linear functional is surjective.
  • Indexing into a vector: The second element of a three-vector is given by the one-form That is, the second element of is
  • Mean: The mean element of an -vector is given by the one-form That is,
  • 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 given by the one-form where is the discount rate. That is,

    Linear functionals in R''n''

Suppose that vectors in the real coordinate space are represented as column vectors
For each row vector there is a linear functional defined by
and each linear functional can be expressed in this form.
This can be interpreted as either the matrix product or the dot product of the row vector and the column vector :

Trace of a square matrix

The trace of a square matrix 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 from the set of all matrices. The trace is a linear functional on this space because and for all scalars and all matrices

(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
is a linear functional from the vector space of continuous functions on the interval to the real numbers. The linearity of follows from the standard facts about the integral:

Evaluation

Let denote the vector space of real-valued polynomial functions of degree defined on an interval If then let be the evaluation functional
The mapping is linear since
If are distinct points in then the evaluation functionals form a basis of the dual space of .

Non-example

A function having the equation of a line with is a linear functional on, 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

If are distinct points in, then the linear functionals defined above form a basis of the dual space of, the space of polynomials of degree The integration functional is also a linear functional on, and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients for which
for all 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 functions.

Dual vectors and bilinear forms

Every non-degenerate bilinear form on a finite-dimensional vector space induces an isomorphism such that
where the bilinear form on is denoted .
The inverse isomorphism is, where is the unique element of such that
for all
The above defined vector is said to be the dual vector of
In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem. There is a mapping from into its V.

Relationship to bases

Basis of the dual space

Let the vector space have a basis, not necessarily orthogonal. Then the dual space has a basis called the dual basis defined by the special property that
Or, more succinctly,
where is the Kronecker delta. Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.
A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ,
Then, applying the functional to a basis vector yields
due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
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 have basis In three dimensions, the dual basis can be written explicitly
for where ε is the Levi-Civita symbol and the inner product on.
In higher dimensions, this generalizes as follows
where is the Hodge star operator.

Over a ring

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

Suppose that is a vector space over Restricting scalar multiplication to gives rise to a real vector space called the of
Any vector space over is also a vector space over endowed with a complex structure; that is, there exists a real vector subspace such that we can write as -vector spaces.

Real versus complex linear functionals

Every linear functional on is complex-valued while every linear functional on is real-valued. If then a linear functional on either one of or is non-trivial if and only if it is surjective, where the image of a linear functional on is while the image of a linear functional on is
Consequently, the only function on that is both a linear functional on and a linear function on is the trivial functional; in other words, where denotes the space's algebraic dual space.
However, every -linear functional on is an -linear , but unless it is identically it is not an -linear on because its range is 2-dimensional over Conversely, a non-zero -linear functional has range too small to be a -linear functional as well.

Real and imaginary parts

If then denote its real part by and its imaginary part by
Then and are linear functionals on and
The fact that for all implies that for all
and consequently, that and
The assignment defines a bijective -linear operator whose inverse is the map defined by the assignment that sends to the linear functional defined by
The real part of is and the bijection is an -linear operator, meaning that and for all and
Similarly for the imaginary part, the assignment induces an -linear bijection whose inverse is the map defined by sending to the linear functional on defined by
This relationship was discovered by Henry Löwig in 1934, 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

Suppose is a linear functional on with real part and imaginary part
Then if and only if if and only if
Assume that is a topological vector space. Then is continuous if and only if its real part is continuous, if and only if 's imaginary part is continuous. That is, either all three of and are continuous or none are continuous. This remains true if the word "continuous" is replaced with the word "bounded". In particular, if and only if where the prime denotes the space's continuous dual space.
Let If for all scalars of unit length then
Similarly, if denotes the complex part of then implies
If is a normed space with norm and if is the closed unit ball then the supremums above are the operator norms of and so that
This conclusion extends to the analogous statement for polars of balanced sets in general topological vector spaces.
  • If is a complex Hilbert space with a inner product that is antilinear in its first coordinate then becomes a real Hilbert space when endowed with the real part of Explicitly, this real inner product on is defined by for all and it induces the same norm on as because for all vectors Applying the Riesz representation theorem to guarantees the existence of a unique vector such that for all vectors The theorem also guarantees that and It is readily verified that Now and the previous equalities imply that 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
If is a topological vector space, the space of continuous linear functionals — the — is often simply called the dual space. If is a Banach space, then so is its 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 topological vector space is continuous if and only if there exists a continuous seminorm on such that