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 V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted Hom(V, k), or, when the field k is understood, ; other notations are also used, such as , or  When vectors are represented by column vectors (as it 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).
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 (i.e. its range is all of k).
Linear functionals in Rn
Suppose that vectors in the real coordinate space are represented as column vectors
For each row vector there is a linear functional defined by
This can be interpreted as either the matrix product or the dot product of the row vector and the column vector :
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 [a, b], to the real numbers. The linearity of I follows from the standard facts about the integral:
Let Pn denote the vector space of real-valued polynomial functions of degree defined on an interval [a, b]. If then let be the evaluation functional
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 Misner, Thorne & Wheeler (1973).
Application to quadrature
If are distinct points in [a, b], then the linear functionals defined above form a basis of the dual space of Pn, the space of polynomials of degree The integration functional I is also a linear functional on Pn, and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients for which
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.
Dual vectors and bilinear forms
The inverse isomorphism is V∗ → V : v∗ ↦ v, where v is the unique element of V such that
The above defined vector v∗ ∈ V∗ is said to be the dual vector of
Relationship to bases
Basis of the dual space
Or, more succinctly,
A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ("components") ui,
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 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
In higher dimensions, this generalizes as follows
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 M over a ring R, a linear form on M is a linear map from M to R, where the latter is considered as a module over itself. The space of linear forms is always denoted Homk(V, k), whether k is a field or not. It is an right module, if V is a left module.
Change of field
Suppose that is a vector space over Restricting scalar multiplication to gives rise to a real vector space called the realification 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 (formally) write as -vector spaces.
Every linear functional on (respectively, on ) is complex-valued (resp. real-valued) and it is non-trivial (i.e. not identically ) if and only if it is surjective (because if then for any scalar ), in which case its image is (resp. 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 operator (meaning that it is additive and homogeneous over ), but unless it is identically it is not an -linear functional on because its range (which is ) is 2-dimensional over Conversely, a non-zero -linear functional has range too small to be a -linear functional as well.
The assignment defines a bijective -linear operator whose inverse is the map defined by the assignment that sends to the linear functional defined by
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.
Suppose is a linear functional on with real part and imaginary part
- 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. 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 (meaning ) then[proof 1] If thenwhere denotes the complex part of In particular, if is a normed space thenwhere all operator norms are defined in the usual way as supremums of absolute values over the closed unit ball 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 (complex) inner product that is antilinear in its first coordinate (and linear in the second) 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 (resp. to ) guarantees the existence of a unique vector (resp. ) such that (resp. ) 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
If is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If 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 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 dimensions the continuous dual is a proper subspace of the algebraic dual.
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
A vector subspace of is called maximal if (meaning and ) and does not exist a vector subspace of such that A vector subspace of is maximal if and only if it is the kernel of some non-trivial linear functional on (that is, for some linear functional on that is not identically 0). An affine hyperplane in is a translate of a maximal vector subspace. By linearity, a subset of is a affine hyperplane if and only if there exists some non-trivial linear functional on such that  If is a linear functional and is a scalar then This equality can be used to relate different level sets of Moreover, if then the kernel of can be reconstructed from the affine hyperplane by
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.
- f can be written as a linear combination of ; that is, there exist scalars such that ;
- there exists a real number r such that for all and all
Any (algebraic) linear functional on a vector subspace can be extended to the whole space; for example, the evaluation functionals described above can be extended to the vector space of polynomials on all of 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,
Hahn–Banach dominated extension theorem(Rudin 1991, Th. 3.2) †�� If is a sublinear function, and is a linear functional on a linear subspace which is dominated by p on M, then there exists a linear extension of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that
Equicontinuity of families of linear functionals
For any subset H of the following are equivalent:
- H is equicontinuous;
- H is contained in the polar of some neighborhood of in X;
- the (pre)polar of H is a neighborhood of in X;
If H 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 (and thus that every equicontinuous subset weak-* relatively compact).
- Discontinuous linear map
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Positive linear functional
- Multilinear form – Map from multiple vectors to an underlying field of scalars, linear in each argument
- Topological vector space – Vector space with a notion of nearness
- For instance,
- It is true if so assume otherwise. Since for all scalars it follows that If then let and be such that and where if then take Then and because is a real number, By assumption so Since was arbitrary, it follows that
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