Editing Dirac delta function (section)

====Resolutions of the identity====
Given a complete [[orthonormal basis]] set of functions {{math|{{brace|''φ''<sub>''n''</sub>}}}} in a separable Hilbert space, for example, the normalized [[eigenvector]]s of a [[Compact operator on Hilbert space#Spectral theorem|compact self-adjoint operator]], any vector {{mvar|f}} can be expressed as
<math display="block">f = \sum_{n=1}^\infty \alpha_n \varphi_n. </math>
The coefficients {α<sub>n</sub>} are found as
<math display="block">\alpha_n = \langle \varphi_n, f \rangle,</math>
which may be represented by the notation:
<math display="block">\alpha_n =  \varphi_n^\dagger f, </math>
a form of the [[bra–ket notation]] of Dirac.<ref>

The development of this section in bra–ket notation is found in {{harv|Levin|2002|loc= Coordinate-space wave functions and completeness, pp.=109''ff''}}</ref> Adopting this notation, the expansion of {{mvar|f}} takes the [[Dyadic tensor|dyadic]] form:{{sfn|Davis|Thomson|2000|loc=Perfect operators, p.344}}

<math display="block">f =  \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). </math>

Letting {{mvar|I}} denote the [[identity operator]] on the Hilbert space, the expression

<math display="block">I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, </math>

is called a [[Borel functional calculus#Resolution of the identity|resolution of the identity]]. When the Hilbert space is the space {{math|''L''<sup>2</sup>(''D'')}} of square-integrable functions on a domain {{mvar|D}}, the quantity:

<math display="block">\varphi_n \varphi_n^\dagger, </math>

is an integral operator, and the expression for {{mvar|f}} can be rewritten

<math display="block">f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.</math>

The right-hand side converges to {{mvar|f}} in the {{math|''L''<sup>2</sup>}} sense.  It need not hold in a pointwise sense, even when {{mvar|f}} is a continuous function.  Nevertheless, it is common to abuse notation and write

<math display="block">f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, </math>

resulting in the representation of the delta function:{{sfn|Davis|Thomson|2000|loc=Equation 8.9.11, p. 344}}

<math display="block">\delta(x-\xi) = \sum_{n=1}^\infty  \varphi_n (x) \varphi_n^*(\xi). </math>

With a suitable [[rigged Hilbert space]] {{math|(Φ, ''L''<sup>2</sup>(''D''), Φ*)}} where {{math|Φ ⊂ ''L''<sup>2</sup>(''D'')}} contains all compactly supported smooth functions, this summation may converge in {{math|Φ*}}, depending on the properties of the basis {{math|''φ''<sub>''n''</sub>}}.  In most cases of practical interest, the orthonormal basis comes from an integral or differential operator (e.g. the [[heat kernel]]), in which case the series converges in the [[Distribution (mathematics)#Distributions|distribution]] sense.{{sfn|de la Madrid|Bohm|Gadella|2002}}