Linear form

Summary

In mathematics, a linear form (also known as a linear functional,[1] 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),[2] or, when the field k is understood, ;[3] other notations are also used, such as ,[4][5] or [2] 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 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

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 (Lax (1996) proves this last fact using Lagrange interpolation).

Non-example

A function having the equation of a line with (for example, ) is not a linear functional on , since it is not linear.[nb 1] It is, however, affine-linear.

Visualization

Geometric interpretation of a 1-form α as a stack of hyperplanes of constant value, each corresponding to those vectors that α maps to a given scalar value shown next to it along with the "sense" of increase. The   zero plane is through the origin.

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).

Applications

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

for all This forms the foundation of the theory of numerical quadrature.[6]

In quantum mechanics

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

Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.[7]

Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism VV : vv such that

where the bilinear form on V is denoted (for instance, in Euclidean space, is the dot product of v and w).

The inverse isomorphism is VV : vv, where v is the unique element of V such that

for all

The above defined vector vV 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 VV from V into its continuous dual space V.

Relationship to bases

Basis of the dual space

Let the vector space V 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 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 ("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

for where ε is the Levi-Civita symbol and the inner product (or dot product) on V.

In higher dimensions, this generalizes as follows

where 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 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.

The existence of "enough" linear forms on a module is equivalent to projectivity.[8]

Dual Basis Lemma — An R-module M is projective if and only if there exists a subset and linear forms such that, for every only finitely many are nonzero, and

Change of field

Suppose that is a vector space over Restricting scalar multiplication to gives rise to a real vector space[9] 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.

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 (meaning not identically ) if and only if it is surjective (because if then for any scalar ), 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 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.

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 [9]

and consequently, that and [10]

The assignment defines a bijective[10] -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 [10] 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 (although it is usually credited to F. Murray),[11] 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.[9]

Let If for all scalars of unit length (meaning ) then[proof 1][12]

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 (defined in the usual way) of and so that [12]
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

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 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.

A linear functional f on a (not necessarily locally convex) topological vector space X is continuous if and only if there exists a continuous seminorm p on X such that [13]

Characterizing closed subspaces

Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed,[14] and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.[15]

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 [11] 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.

Theorem[16][17] — If are linear functionals on X, then the following are equivalent:

  1. f can be written as a linear combination of ; that is, there exist scalars such that ;
  2. ;
  3. there exists a real number r such that for all and all

If f is a non-trivial linear functional on X with kernel N, satisfies and U is a balanced subset of X, then if and only if for all [15]

Hahn–Banach theorem

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[18](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

for all and
for all

Equicontinuity of families of linear functionals

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

For any subset H of the following are equivalent:[19]

  1. H is equicontinuous;
  2. H is contained in the polar of some neighborhood of in X;
  3. 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.[19] 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).[20][19]

See also

Notes

Footnotes

  1. ^ For instance,

Proofs

  1. ^ 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

References

  1. ^ Axler (2015) p. 101, §3.92
  2. ^ a b Tu (2011) p. 19, §3.1
  3. ^ Katznelson & Katznelson (2008) p. 37, §2.1.3
  4. ^ Axler (2015) p. 101, §3.94
  5. ^ Halmos (1974) p. 20, §13
  6. ^ Lax 1996
  7. ^ Misner, Thorne & Wheeler (1973) p. 57
  8. ^ Clark, Pete L. Commutative Algebra (PDF). Unpublished. Lemma 3.12.
  9. ^ a b c Rudin 1991, pp. 57.
  10. ^ a b c Narici & Beckenstein 2011, pp. 9–11.
  11. ^ a b Narici & Beckenstein 2011, pp. 10–11.
  12. ^ a b Narici & Beckenstein 2011, pp. 126–128.
  13. ^ Narici & Beckenstein 2011, p. 126.
  14. ^ Rudin 1991, Theorem 1.18
  15. ^ a b Narici & Beckenstein 2011, p. 128.
  16. ^ Rudin 1991, pp. 63–64.
  17. ^ Narici & Beckenstein 2011, pp. 1–18.
  18. ^ Narici & Beckenstein 2011, pp. 177–220.
  19. ^ a b c Narici & Beckenstein 2011, pp. 225–273.
  20. ^ Schaefer & Wolff 1999, Corollary 4.3.

Bibliography