One important application of the spectral theorem is to define a "functional calculus." That is to say, if formula_116 is a function on the real line and formula_117 is a self-adjoint operator, we wish to define the operator formula_118. If formula_117 has a true orthonormal basis of eigenvectors formula_120 with eigenvalues formula_121, then formula_118 is the operator with eigenvectors formula_120 and eigenvalues formula_124. The goal of functional calculus is to extend this idea to the case where formula_117 has continuous spectrum.