Editing Linear form (section)

== Applications ==

=== Application to quadrature ===
If <math>x_0, \ldots, x_n</math> are <math>n + 1</math> distinct points in {{closed-closed|''a'', ''b''}}, then the linear functionals <math>\operatorname{ev}_{x_i} : f \mapsto f\left(x_i\right)</math> defined above form a [[Basis of a vector space|basis]] of the dual space of {{math|''P<sub>n</sub>''}}, the space of polynomials of degree <math>\leq n.</math>  The integration functional {{math|''I''}} is also a linear functional on {{math|''P<sub>n</sub>''}}, and so can be expressed as a linear combination of these basis elements.  In symbols, there are coefficients <math>a_0, \ldots, a_n</math> for which
<math display="block">I(f) = a_0 f(x_0) + a_1 f(x_1) + \dots + a_n f(x_n)</math>
for all <math>f \in P_n.</math> This forms the foundation of the theory of [[numerical quadrature]].<ref>{{harvnb|Lax|1996}}</ref>

=== In quantum mechanics ===
Linear functionals are particularly important in [[quantum mechanics]]. Quantum mechanical systems are represented by [[Hilbert space]]s, which are [[Antilinear|anti]]–[[Linear isomorphism|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]].

=== Distributions ===
In the theory of [[generalized function]]s, certain kinds of generalized functions called [[distribution (mathematics)|distributions]] can be realized as linear functionals on spaces of [[test function]]s.