Linear form

Suppose that vectors in the real coordinate space Rⁿ are represented as column vectors

${\displaystyle {\vec {x}}={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}$

For each row vector [a₁ ⋯ a_n] there is a linear functional f defined by

${\displaystyle f({\vec {x}})=a_{1}x_{1}+\cdots +a_{n}x_{n},}$

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 [a₁ ⋯ a_n] and the column vector ${\displaystyle {\vec {x}}}$:

${\displaystyle f({\vec {x}})={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}$

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

${\displaystyle I(f)=\int _{a}^{b}f(x)\,dx}$

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:

${\displaystyle {\begin{aligned}I(f+g)&=\int _{a}^{b}[f(x)+g(x)]\,dx=\int _{a}^{b}f(x)\,dx+\int _{a}^{b}g(x)\,dx=I(f)+I(g)\\I(\alpha f)&=\int _{a}^{b}\alpha f(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx=\alpha I(f).\end{aligned}}}$

Let P_n denote the vector space of real-valued polynomial functions of degree ≤n defined on an interval [a, b].  If c ∈ [a, b], then let ev_c : P_n → R be the evaluation functional

${\displaystyle \operatorname {ev} _{c}f=f(c).}$

The mapping f → f(c) is linear since

${\displaystyle {\begin{aligned}(f+g)(c)&=f(c)+g(c)\\(\alpha f)(c)&=\alpha f(c).\end{aligned}}}$

If x₀, ..., x_n are n + 1 distinct points in [a, b], then the evaluation functionals ev_xi, i = 0, 1, ..., n form a basis of the dual space of P_n.  (Lax (1996) proves this last fact using Lagrange interpolation.)

A function f having the equation of a line f(x) = a + rx with a ≠ 0 (e.g. f(x) = 1 + 2x) is not a linear functional on ℝ, since it is not linear.[nb 1] It is, however, affine-linear.

If x₀, ..., x_n are n + 1 distinct points in [a, b], then the linear functionals ev_xi : f → f(x_i) defined above form a basis of the dual space of P_n, the space of polynomials of degree ≤ n. The integration functional I is also a linear functional on P_n, and so can be expressed as a linear combination of these basis elements. In symbols, there are coefficients a₀, ..., a_n for which

${\displaystyle I(f)=a_{0}f(x_{0})+a_{1}f(x_{1})+\dots +a_{n}f(x_{n})}$

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

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.

Let the vector space V have a basis ${\displaystyle {\vec {e}}_{1},{\vec {e}}_{2},\dots ,{\vec {e}}_{n}}$, not necessarily orthogonal.  Then the dual space V* has a basis ${\displaystyle {\tilde {\omega }}^{1},{\tilde {\omega }}^{2},\dots ,{\tilde {\omega }}^{n}}$ called the dual basis defined by the special property that

${\displaystyle {\tilde {\omega }}^{i}({\vec {e}}_{j})=\left\{{\begin{matrix}1&\mathrm {if} \ i=j\\0&\mathrm {if} \ i\not =j.\end{matrix}}\right.}$

Or, more succinctly,

${\displaystyle {\tilde {\omega }}^{i}({\vec {e}}_{j})=\delta _{ij}}$

where δ is the Kronecker delta.  Here the superscripts of the basis functionals are not exponents but are instead contravariant indices.

A linear functional ${\displaystyle {\tilde {u}}}$ belonging to the dual space ${\displaystyle {\tilde {V}}}$ can be expressed as a linear combination of basis functionals, with coefficients ("components") u_i,

${\displaystyle {\tilde {u}}=\sum _{i=1}^{n}u_{i}\,{\tilde {\omega }}^{i}.}$

Then, applying the functional ${\displaystyle {\tilde {u}}}$ to a basis vector e_j yields

${\displaystyle {\tilde {u}}({\vec {e}}_{j})=\sum _{i=1}^{n}\left(u_{i}\,{\tilde {\omega }}^{i}\right){\vec {e}}_{j}=\sum _{i}u_{i}\left[{\tilde {\omega }}^{i}\left({\vec {e}}_{j}\right)\right]}$

due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals.  Then

${\displaystyle {\begin{aligned}{\tilde {u}}({\vec {e}}_{j})&=\sum _{i}u_{i}\left[{\tilde {\omega }}^{i}\left({\vec {e}}_{j}\right)\right]=\sum _{i}u_{i}{\delta ^{i}}_{j}\\&=u_{j}.\end{aligned}}}$

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 ${\displaystyle {\vec {e}}_{1},\dots ,{\vec {e}}_{n}}$.  In three dimensions (n = 3), the dual basis can be written explicitly

${\displaystyle {\tilde {\omega }}^{i}({\vec {v}})={1 \over 2}\,\left\langle {\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon ^{ijk}\,({\vec {e}}_{j}\times {\vec {e}}_{k}) \over {\vec {e}}_{1}\cdot {\vec {e}}_{2}\times {\vec {e}}_{3}},{\vec {v}}\right\rangle ,}$

for i = 1, 2, 3, where ε is the Levi-Civita symbol and ${\displaystyle \langle ,\rangle }$ the inner product (or dot product) on V.

In higher dimensions, this generalizes as follows

${\displaystyle {\tilde {\omega }}^{i}({\vec {v}})=\left\langle {\frac {{\underset {{}^{1\leq i_{2}<i_{3}<\dots <i_{n}\leq n}}{\sum }}\varepsilon ^{ii_{2}\dots i_{n}}(\star {\vec {e}}_{i_{2}}\wedge \dots \wedge {\vec {e}}_{i_{n}})}{\star ({\vec {e}}_{1}\wedge \dots \wedge {\vec {e}}_{n})}},{\vec {v}}\right\rangle ,}$

where ${\displaystyle \star }$ is the Hodge star operator.

Continuous linear functionals have nice properties for analysis: a linear functional is continuous if and only if its kernel is closed,[11] and a non-trivial continuous linear functional is an open map, even if the (topological) vector space is not complete.[12]

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}.[9]

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, 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.[12]

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,

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

For any subset H of X', the following are equivalent:[16]

1. H is equicontinuous;
2. H is contained in the polar of some neighborhood of 0 in X;
3. 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.[16] 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).[17][16]