Linear form


In linear algebra, a linear form is a linear map from a vector space to its field of scalars. In ℝn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by 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 over a field k, then a linear functional f is a function from V to k that is linear:
The set of all linear functionals from V to k, denoted by Homk, forms a vector space over k with the operations of addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space. It is often written V∗, V′, or Vᐯ when the field k is understood.

Linear functionals on real or complex vector spaces

We assume throughout that all vector spaces that we consider are either vector spaces over the set of real numbers or vector spaces over the set of complex numbers.
We assume that is a vector space over where is either or.

Basic definitions

Observe that if is a vector space over then a "linear functional on " is a linear map of the form , while if is a vector space over then a "linear functional on " is a linear map of the form .
Recall that is a vector space over so we can ask what are the linear functionals on ?
A function is a linear functional on if and only if it is of the form for some real number.
Note in particular that a function having the equation of a line with is not a linear functional on .
It is however, a type of function known as an affine linear functional.
A linear functional is non-trivial if and only if it is surjective.

Relationships with other maps

Relationship between real and complex linear functionals

Suppose that is a vector space over.
Let denote when it is considered as a vector space over.
Note that every linear functional on is, by definition, complex-valued while every linear functional on is real-valued.
If is a real linear functional on then is a linear functional on if and only if is trivial .
Thus, we note the following important technicality:
However, a real linear functional on does induce a canonical linear functional defined by for all, where.
Now suppose that and let denote the real part of so that.
Then for all, and so that
This shows that,, and each completely determine one another and it follows that and are real linear functionals on and the canonical linear functional on induced by is .
Furthermore, for all,
Thus the map, denoted by, defines a one-to-one correspondence from onto whose inverse is the map.
Furthermore, is linear as a map over .
Similarly, the inverse of the surjective map defined by is the map that sends to the linear functional.
This relationship was discovered by Henry Löwig in 1934.
If is a linear functional on a real or complex vector space and if is a seminorm on, then on if and only if on .
;Topological consequences
If is a complex topological vector space, then either all three of,, and are continuous, or else all three are discontinuous.
Moreover, if is a complex normed space then .

Relationships with seminorms and sublinear functions

A sublinear function on a vector space is a function that satisfies the following two properties:

  1. Subadditivity: for all ;
  2. Positive homogeneity: for any positive real and any.
A seminorm on is a sublinear function that satisfies the following additional property:

  1. Absolute homogeneity: for all and all scalars ;
Note that every linear functional on a real vector space is a sublinear function, although there are sublinear functions that are not linear functionals.
Unlike linear functionals, a seminorm is valued in the non-negative real numbers, so the only linear function that is also a seminorm is the trivial 0 map.
However, if is a linear functional on a vector space, then its absolute value is a seminorm on
If is a linear functional on a real vector space and is a seminorm on, then if and only if.

Hahn-Banach theorem

The Hahn-Banach theorem is considered one of the most important results of the subfield of mathematics called functional analysis.
Due to its importance, the Hahn-Banach theorem has been generalized many times and today "Hahn-Banach theorem" refers to any one of a collection of theorems.
The general idea behind a Hahn-Banach theorem is that it gives conditions under which a linear functional on a vector subspace of can be extended to a linear functional on the whole of .
The following is one of many results known collectively as "Hahn-Banach theorems."

Relationships between multiple linear functionals

Any two linear functionals with the same kernel are proportional.
This fact can be generalized to the following theorem.
If is a non-trivial linear functional on with kernel, satisfies, and is a balanced subset of, then if and only if for all.

Hyperplanes and maximal subspaces

A vector subspace of is maximal in if and only if it is the kernel of some non-trivial linear functional on .
A vsubset of is a hyperplane in if and only if there exists some non-trivial linear functional on and some scalar such that, or equivalently, if and only if there exists some non-trivial linear functional on such that.

Continuous linear functionals

is a field of mathematics dedicated to studying vector spaces over or when they are endowed with a topology making addition and scalar multiplication continuous.
Such objects are call topological vector spaces.
Prominent examples of TVSs include Euclidean space, normed spaces, Banach spaces, and Hilbert spaces.
If is a topological vector space over then the continuous dual space or simply the dual space is the vector space over consisting of all continuous linear functionals on.
If is a Banach space, then so is its dual space.
To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual space.
In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensional locally convex space, 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.
Every non-trivial continuous linear functional on a TVS is an open map.
A linear functional on a complex TVS is bounded if and only if its real part is bounded.
A linear functional is continuous if and only if its kernel is closed.
If is a linear functional on a topological vector space and if is a continuous sublinear function on then || ≤ pmath|f

Equicontinuity of families of linear functionals

Let be a topological vector space with continuous dual space.
For any subset of, the following are equivalent:

  1. is equicontinuous;
  2. is contained in the polar of some neighborhood of in ;
  3. the polar of is a neighborhood of 0 in ;
If is an equicontinuous subset of 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 implies that the weak-* closure of an equicontinuous subset of is weak-* compact.

Examples and applications

Linear functionals in R''n''

Suppose that vectors in the real coordinate space Rn are represented as column vectors
For each row vector there is a linear functional f 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 :

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 C of continuous functions on the interval to the real numbers. The linearity of follows from the standard facts about the integral:

Evaluation

Let Pn denote the vector space of real-valued polynomial functions of degree ≤n defined on an interval . If c ∈ , then let be the evaluation functional
The mapping f → f is linear since
If x0,..., xn are distinct points in, then the evaluation functionals form a basis of the dual space of Pn.

Application to quadrature

The integration functional defined above defines a linear functional on the subspace of polynomials of degree. If are distinct points in, then there are coefficients for which
for all. This forms the foundation of the theory of numerical quadrature.
This follows from the fact that the linear functionals defined above form a basis of the dual space of.

Linear functionals 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.

Visualizing linear functionals

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.

Dual vectors and bilinear forms

Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism such that
where the bilinear form on V is denoted .
The inverse isomorphism is, where v is the unique element of V such that
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 into the continuous dual space V. However, this mapping is antilinear rather than linear.

Bases in finite dimensions

Basis of the dual space in finite dimensions

Let the vector space V have a basis, not necessarily orthogonal. Then the dual space 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 indices.
A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients 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
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 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 basis. In three dimensions, the dual basis can be written explicitly
for i = 1, 2, 3, where ε is the Levi-Civita symbol and the inner product on V.
In higher dimensions, this generalizes as follows
where is the Hodge star operator.