Linear Form

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## Examples

### Linear functionals in R^{n}

### (Definite) Integration

### Evaluation

### Non-example

## Visualization

## Applications

### Application to quadrature

### In quantum mechanics

### Distributions

## Dual vectors and bilinear forms

## Relationship to bases

### Basis of the dual space

### The dual basis and inner product

## Change of field

## In infinite dimensions

### Characterizing closed subspaces

#### Hyperplanes and maximal subspaces

#### Relationships between multiple linear functionals

### Hahn-Banach theorem

### Equicontinuity of families of linear functionals

## See also

## Notes

## References

## Bibliography

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Linear Form

In linear algebra, 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. If vectors are represented as column vectors (as is the Wikipedia convention), 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:

- for all
- for all

The set of all linear functionals from *V* to *k*, denoted by Hom_{k}(*V*,*k*), 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?*, *V*^{#} or *V*^{∨} when the field *k* is understood.

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).

Suppose that vectors in the real coordinate space **R**^{n} are represented as column vectors

For each row vector [*a*_{1} ... *a*_{n}] 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 [*a*_{1} ... *a*_{n}] 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 C[*a*, *b*] 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 *P _{n}* denote the vector space of real-valued polynomial functions of degree n defined on an interval [

The mapping *f* -> *f*(*c*) is linear since

If *x*_{0}, ..., *x _{n}* are distinct points in , then the evaluation functionals form a basis of the dual space of

A function f having the equation of a line *f*(*x*) = *a* + *rx* with *a* ? 0 (e.g. *f*(*x*) = 1 + 2*x*) is *not* a linear functional on R, since it is not linear.^{[nb 1]} 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).

If *x*_{0}, ..., *x*_{n} are *n* + 1 distinct points in [*a*, *b*], then the linear functionals ev* _{xi}* :

for all *f* ? *P*_{n}. This forms the foundation of the theory of numerical quadrature.^{[1]}

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.

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 (for instance, in Euclidean space is the dot product of *v* and *w*).

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*^{*}.

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 ? 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") *u _{i}*,

Then, applying the functional to a basis vector *e _{j}* 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.

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 , the dual basis can be written explicitly

for *i* = 1, 2, 3, where *?* 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.

Any vector space X over C is also a vector space over R, endowed with a complex structure; that is, there exists a real vector subspace *X*_{R} such that we can (formally) write *X* = *X*_{R} ? *X*_{R}*i* as R-vector spaces. Every C-linear functional on X is a R-linear operator, but it is not an R-linear *functional* on X, because its range (namely, C) is 2-dimensional over R. (Conversely, a R-linear functional has range too small to be a C-linear functional as well.)

However, every C-linear functional uniquely determines an R-linear functional on *X*_{R} by restriction. More surprisingly, this result can be reversed: every R-linear functional g on X induces a canonical C-linear functional *L*_{g} ? *X*^{#}, such that the real part of *L _{g}* is g: define

*L*_{g}(*x*) :=*g*(*x*) -*i**g*(*ix*) for all*x*?*X*.

*L*_{ •} is R-linear (i.e. *L*_{g+h} = *L*_{g} + *L*_{h} and *L*_{rg} = *r* *L*_{g} for all *r* ? R and *g*, *h* ? *X*_{R}^{#}). Similarly, the inverse of the surjection Hom(*X*,ℂ) -> Hom(*X*,R) defined by *f* ? Im *f* is the map *I* ? (*x* ? *I*(*ix*) + *i* *I*(*x*)).

This relationship was discovered by Henry Löwig in 1934 (although it is usually credited to F. Murray),^{[3]} and can be generalized to arbitrary finite extensions of a field in the natural way.

Below, all vector spaces are over either the real numbers R or the complex numbers C.

If *V* is a topological vector space, the space of continuous linear functionals -- the *continuous dual* -- is often simply called the dual space. If *V* 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 || p.^{[4]}

Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed,^{[5]} and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.^{[6]}

A vector subspace M of X is called **maximal** if *M* ⊊ *X*, but there are no vector subspaces N satisfying *M* ⊊ *N* ⊊ *X*. M is maximal if and only if it is the kernel of some non-trivial linear functional on X (i.e. *M* = ker *f* for some non-trivial linear functional f on X). A **hyperplane** in X is a translate of a maximal vector subspace. By linearity, a subset H of X is a hyperplane if and only if there exists some non-trivial linear functional f on X such that *H* = { *x* ? *X* : *f*(*x*) = 1}.^{[3]}

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 ^{[7]}^{[8]}** — If

- f can be written as a linear combination of
*g*_{1}, ...,*g*_{n}(i.e. there exist scalars*s*_{1}, ...,*s*_{n}such that*f*=*s*_{1}*g*_{1}+ +*s*_{n}*g*_{n}); - ?
^{n}_{i=1}Ker*g*_{i}? Ker*f*; - there exists a real number r such that || r || for all
*x*?*X*and all i.

If f is a non-trivial linear functional on X with kernel N, *x* ? *X* satisfies *f*(*x*) = 1, and U is a balanced subset of X, then *N* ∩ (*x* + *U*) = ? if and only if |*f*(*u*)| < 1 for all *u* ? *U*.^{[6]}

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 R. 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 ^{[9]}(Rudin 1991, Th. 3.2)** — If

*F*(*m*) =*f*(*m*) for all*m*?*M*,- || p(
*x*) for all*x*?*X*.

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

For any subset *H* of *X*, the following are equivalent:^{[10]}

*H*is equicontinuous;*H*is contained in the polar of some neighborhood of 0 in X;- the (pre)polar of
*H*is a neighborhood of 0 in X;

If *H* is an equicontinuous subset of *X* then the following sets are also equicontinuous:
the weak-* closure, the balanced hull, the convex hull, and the convex balanced hull.^{[10]}
Moreover, Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of *X* is weak-* compact (and thus that every equicontinuous subset weak-* relatively compact).^{[11]}^{[10]}

- 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,*f*(1 + 1) =*a*+ 2*r*? 2*a*+ 2*r*=*f*(1) +*f*(1).

**^**Lax 1996**^**J.A. Wheeler; C. Misner; K.S. Thorne (1973).*Gravitation*. W.H. Freeman & Co. p. 57. ISBN 0-7167-0344-0.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 10-11. **^**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.

- Bishop, Richard; Goldberg, Samuel (1980), "Chapter 4",
*Tensor Analysis on Manifolds*, Dover Publications, ISBN 0-486-64039-6 - Conway, John B. (1990).
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*Linear operators*(in Romanian). New York: Interscience Publishers. ISBN 0-471-60848-3. OCLC 18412261. - Halmos, Paul (1974),
*Finite dimensional vector spaces*, Springer, ISBN 0-387-90093-4 - 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.**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.**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. - Wilansky, Albert (2013).
*Modern Methods in Topological Vector Spaces*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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