Editing Spectral theorem (section)

===Direct integrals===
There is also a formulation of the spectral theorem in terms of [[Direct integral|direct integrals]]. It is similar to the multiplication-operator formulation, but more canonical.

Let <math>A</math> be a bounded self-adjoint operator and let <math>\sigma (A)</math> be the spectrum of <math>A</math>. The direct-integral formulation of the spectral theorem associates two quantities to <math>A</math>. First, a measure <math>\mu</math> on <math>\sigma (A)</math>, and second, a family of Hilbert spaces <math>\{H_{\lambda}\},\,\,\lambda\in\sigma (A).</math> We then form the direct integral Hilbert space
<math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda). </math>
The elements of this space are functions (or "sections") <math>s(\lambda),\,\,\lambda\in\sigma(A),</math> such that <math>s(\lambda)\in H_{\lambda}</math> for all <math>\lambda</math>. 
The direct-integral version of the spectral theorem may be expressed as follows:<ref>{{harvnb|Hall|2013}} Theorem 7.19</ref>
{{math theorem|math_statement= If <math>A</math> is a bounded self-adjoint operator, then <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda) </math>
for some measure <math>\mu</math> and some family <math>\{H_{\lambda}\}</math> of Hilbert spaces. The measure <math>\mu</math> is uniquely determined by <math>A</math> up to measure-theoretic equivalence; that is, any two measure associated to the same <math>A</math> have the same sets of measure zero. The dimensions of the Hilbert spaces <math>H_{\lambda}</math> are uniquely determined by <math>A</math> up to a set of <math>\mu</math>-measure zero.}}

The spaces <math>H_{\lambda}</math> can be thought of as something like "eigenspaces" for <math>A</math>. Note, however, that unless the one-element set <math>\lambda</math> has positive measure, the space <math>H_{\lambda}</math> is not actually a subspace of the direct integral. Thus, the <math>H_{\lambda}</math>'s should be thought of as "generalized eigenspace"—that is, the elements of <math>H_{\lambda}</math> are "eigenvectors" that do not actually belong to the Hilbert space.

Although both the multiplication-operator and direct integral formulations of the spectral theorem express a self-adjoint operator as unitarily equivalent to a multiplication operator, the direct integral approach is more canonical. First, the set over which the direct integral takes place (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function <math>\lambda\mapsto\lambda</math>.