Editing Spectral theorem (section)

===Multiplication operator version===
An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator, a relatively simple type of operator.{{math theorem
| math_statement = Let <math>A</math>  be a bounded self-adjoint operator on a Hilbert space <math> V </math>.  Then there is a [[measure space]] <math>(X, \Sigma, \mu) </math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>\lambda </math> on <math>X</math> and a [[unitary operator]] <math>U : V \to L^2(X, \mu)</math> such that
<math display="block"> U^* T U = A,</math>
where <math> T </math> is the [[multiplication operator]]:
<math display="block"> [T f](x) = \lambda(x) f(x) </math>
and <math>  \vert  T  \vert </math>  <math> = \vert  \lambda  \vert_\infty </math>.
| name = '''Theorem'''<ref>{{harvnb|Hall|2013}} Theorem 7.20</ref>
}}Multiplication operators are a direct generalization of diagonal matrices. A finite-dimensional Hermitian vector space <math>V</math> may be coordinatized as the space of functions <math>f: B \to \C </math> from a basis <math>B</math> to the complex numbers, so that the <math>B</math>-coordinates of a vector are the values of the corresponding function <math>f</math>. The finite-dimensional spectral theorem for a self-adjoint operator <math>A: V \to V </math> states that there exists an orthonormal basis of eigenvectors <math>B</math>, so that the inner product becomes the [[dot product]] with respect to the <math>B</math>-coordinates: thus <math>V</math> is isomorphic to <math>L^2( B ,\mu )  </math> for the discrete unit measure <math>\mu</math> on <math>B</math>. Also <math>A</math> is unitarily equivalent to the multiplication operator <math>[Tf](v)  =  \lambda(v)  f(v)  </math>, where <math>\lambda(v)</math> is the eigenvalue of <math>v \in B  </math>: that is, <math>A</math> multiplies each <math>B</math>-coordinate by the corresponding eigenvalue <math>\lambda(v)</math>, the action of a diagonal matrix. Finally, the [[operator norm]] <math>|A| = |T| </math> is equal to the magnitude of the largest eigenvector <math>|\lambda|_\infty </math>.

The spectral theorem is the beginning of the vast research area of functional analysis called [[operator theory]]; see also [[spectral measure]].

There is also an analogous spectral theorem for bounded [[Normal operator|normal operators]] on Hilbert spaces. The only difference in the conclusion is that now ''<math>\lambda</math>'' may be complex-valued.