Editing Spectral theory (section)

==Spectral theory briefly==
{{Main|Spectral theorem}}
{{See also|Eigenvalue, eigenvector and eigenspace}}
In [[functional analysis]] and [[linear algebra]] the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist.

This topic is easiest to describe by introducing the [[bra–ket notation]] of [[Paul Dirac|Dirac]] for operators.<ref name= Friedman>{{Cite book |title=Principles and Techniques of Applied Mathematics |author=Bernard Friedman |year=1990 |publisher=Dover Publications |page=26 |isbn=0-486-66444-9 |url=https://books.google.com/books?id=gnQeAQAAIAAJ&q=intitle:applied+intitle:mathematics+inauthor:Friedman |edition=Reprint of 1956 Wiley}}</ref><ref name=Dirac>{{Cite book |title=The principles of quantum mechanics |author=PAM Dirac |edition=4th |isbn=0-19-852011-5 |publisher=Oxford University Press |year=1981 |page=29 ''ff'' |url=https://books.google.com/books?id=XehUpGiM6FIC&pg=PA29}}</ref> As an example, a very particular linear operator ''L'' might be written as a [[dyadic product]]:<ref name=Audretsch>{{Cite book |title=Entangled systems: new directions in quantum physics |author=Jürgen Audretsch |page=5 |chapter-url=https://books.google.com/books?id=8NxIgwAOU6IC&pg=PA5 |chapter=Chapter 1.1.2: Linear operators on the Hilbert space |isbn=978-3-527-40684-5 |publisher=Wiley-VCH |year=2007}}</ref><ref name=Howland>{{Cite book |title=Intermediate dynamics: a linear algebraic approach |url=https://books.google.com/books?id=SepP8-W3M0AC&q=dyad+representation+operator&pg=PA69 |page=69 ''ff'' |author=R. A. Howland |publisher=Birkhäuser |year=2006 |isbn=0-387-28059-6 |edition=2nd}}</ref>

:<math> L = | k_1 \rangle \langle b_1 |, </math>

in terms of the "bra"  ⟨{{mvar|b}}<sub>1</sub>|  and the "ket"  |{{mvar|k}}<sub>1</sub>⟩. A function {{mvar|f}} is described by a ''ket'' as |{{mvar|f}} ⟩. The function {{math|''f''(''x'')}} defined on the coordinates <math>(x_1, x_2, x_3, \dots)</math> is denoted as
:<math> f(x)=\langle x | f\rangle </math>
and the magnitude of ''f'' by
:<math> \|f \|^2 = \langle f| f\rangle =\int \langle f| x\rangle \langle x | f \rangle \, dx = \int f^*(x) f(x) \, dx </math>
where the notation (*) denotes a [[complex conjugate]]. This [[inner product]] choice defines a very specific [[inner product space]], restricting the generality of the arguments that follow.<ref name=Lorch2/>

The effect of ''L'' upon a function ''f'' is then described as:

:<math> L | f\rangle = | k_1 \rangle \langle b_1 | f \rangle </math>

expressing the result that the effect of ''L'' on ''f'' is to produce a new function <math> | k_1 \rangle </math> multiplied by the inner product represented by <math>\langle b_1 | f \rangle </math>.
 
A more general linear operator ''L'' might be expressed as:

:<math> L = \lambda_1 | e_1\rangle\langle f_1| +  \lambda_2 | e_2\rangle \langle f_2| +   \lambda_3 | e_3\rangle\langle f_3| + \dots , </math>

where the <math> \{ \, \lambda_i \, \}</math> are scalars and the <math> \{ \, | e_i \rangle \, \} </math> are a [[Basis (linear algebra)|basis]] and the <math> \{ \, \langle f_i | \, \} </math> a [[Dual basis|reciprocal basis]] for the space. The relation between the basis and the reciprocal basis is described, in part, by:

:<math> \langle f_i | e_j \rangle = \delta_{ij} </math>

If such a formalism applies, the <math> \{ \, \lambda_i \, \}</math> are [[eigenvalues]] of ''L'' and the functions <math> \{ \, | e_i \rangle \, \} </math> are [[eigenfunctions]] of ''L''. The eigenvalues are in the ''spectrum'' of ''L''.<ref name= Friedman2>{{Cite book |title=op. cit. |author=Bernard Friedman |year=1990 |page=57 |chapter=Chapter 2: Spectral theory of operators |publisher=Dover Publications |isbn=0-486-66444-9 |url=https://books.google.com/books?id=gnQeAQAAIAAJ&q=intitle:applied+intitle:mathematics+inauthor:Friedman}}</ref>

Some natural questions are: under what circumstances does this formalism work, and for what operators ''L'' are expansions in series of other operators like this possible? Can any function ''f'' be expressed in terms of the eigenfunctions (are they a [[Schauder basis]]) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in [[functional analysis]] and [[Matrix (mathematics)|matrix algebra]].