Linear form - Wikipedia
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In mathematics, a linear form (also known as a linear functional,[1] a one-form, or a covector) is a linear map[nb 1] 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).
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
[edit]
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
[edit]
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
[edit]
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:
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).
A function having the equation of a line
with
(for example,
) is not a linear functional on
, since it is not linear.[nb 2] It is, however, affine-linear.

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
[edit]
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
[edit]
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.
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
[edit]

Every non-degenerate bilinear form on a finite-dimensional vector space V induces an isomorphism V → V∗ : v ↦ v∗ 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 V∗ → V : v∗ ↦ v, where v is the unique element of V such that
for all
The above defined vector v∗ ∈ V∗ 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 V ↦ V∗ from V into its continuous dual space V∗.
Relationship to bases
[edit]
Below, we assume that the dimension is finite. For a discussion of analogous results in infinite dimensions, see Schauder basis.
Basis of the dual space
[edit]
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 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
[edit]
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.
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 a 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
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
[edit]
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
[edit]
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
[edit]
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.
In infinite dimensions
[edit]
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
[edit]
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
[edit]
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
[edit]
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.
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
[edit]
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
[edit]
Let X be a topological vector space (TVS) with continuous dual space
For any subset H of the following are equivalent:[19]
- 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.[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]
- Discontinuous linear map
- Locally convex topological vector space – Vector space with a topology defined by convex open sets
- Positive linear functional – ordered vector space with a partial order
- 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
- ^ Axler (2015) p. 101, §3.92
- ^ a b Tu (2011) p. 19, §3.1
- ^ Katznelson & Katznelson (2008) p. 37, §2.1.3
- ^ Axler (2015) p. 101, §3.94
- ^ Halmos (1974) p. 20, §13
- ^ Lax 1996
- ^ Misner, Thorne & Wheeler (1973) p. 57
- ^ Clark, Pete L. Commutative Algebra (PDF). Unpublished. Lemma 3.12.
- ^ a b c Rudin 1991, pp. 57.
- ^ a b c Narici & Beckenstein 2011, pp. 9–11.
- ^ a b Narici & Beckenstein 2011, pp. 10–11.
- ^ a b Narici & Beckenstein 2011, pp. 126–128.
- ^ Narici & Beckenstein 2011, p. 126.
- ^ Rudin 1991, Theorem 1.18
- ^ a b Narici & Beckenstein 2011, p. 128.
- ^ Rudin 1991, pp. 63–64.
- ^ Narici & Beckenstein 2011, pp. 1–18.
- ^ Narici & Beckenstein 2011, pp. 177–220.
- ^ a b c Narici & Beckenstein 2011, pp. 225–273.
- ^ Schaefer & Wolff 1999, Corollary 4.3.
- Axler, Sheldon (2015), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer, ISBN 978-3-319-11079-0
- Bishop, Richard; Goldberg, Samuel (1980), "Chapter 4", Tensor Analysis on Manifolds, Dover Publications, ISBN 0-486-64039-6
- Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
- Dunford, Nelson (1988). Linear operators (in Romanian). New York: Interscience Publishers. ISBN 0-471-60848-3. OCLC 18412261.
- Halmos, Paul Richard (1974), Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics (1958 2nd ed.), Springer, ISBN 0-387-90093-4
- Katznelson, Yitzhak; Katznelson, Yonatan R. (2008), A (Terse) Introduction to Linear Algebra, American Mathematical Society, ISBN 978-0-8218-4419-9
- Lax, Peter (1996), Linear algebra, Wiley-Interscience, ISBN 978-0-471-11111-5
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schutz, Bernard (1985), "Chapter 3", A first course in general relativity, Cambridge, UK: Cambridge University Press, ISBN 0-521-27703-5
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Tu, Loring W. (2011), An Introduction to Manifolds, Universitext (2nd ed.), Springer, ISBN 978-0-8218-4419-9
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.