Editing Spectral theorem (section)

===Functional calculus===
One important application of the spectral theorem (in whatever form) is the idea of defining a [[functional calculus]]. That is, given a function <math>f</math> defined on the spectrum of <math>A</math>, we wish to define an operator <math>f(A)</math>. If <math>f</math> is simply a positive power, <math>f(x) = x^n</math>, then <math>f(A)</math> is just the <math>n</math>-th power of <math>A</math>, <math>A^n</math>. The interesting cases are where <math>f</math> is a nonpolynomial function such as a square root or an exponential. Either of the versions of the spectral theorem provides such a functional calculus.<ref>E.g., {{harvnb|Hall|2013}} Definition 7.13</ref> In the direct-integral version, for example, <math>f(A)</math> acts as the "multiplication by <math>f</math>" operator in the direct integral:
<math display="block">[f(A)s](\lambda) = f(\lambda) s(\lambda).</math>
That is to say, each space <math>H_{\lambda}</math> in the direct integral is a (generalized) eigenspace for <math>f(A)</math> with eigenvalue <math>f(\lambda)</math>.