Spectral Theory of Operators in Hilbert Space

approximated assign assume bilinear form bounded operators bounded support Cauchy sequence Chapter Clearly closure complete space completely continuous components convergence coordinate system defined denote dense derivative differential operator domain dr(s eigen eigenspace eigenvalue eigenvector equation essentially discrete evidently fact finite number finite support follows formally adjoint formally self-adjoint formula Fourier func function r(s function space functional calculus given hence Hermitean Hilbert space holds ideal elements ideal functions identity implies infinite inner product space integral operator introduce kernel k(s,s Lemma linear combination linear space measure function non-negative notion open interval orthogonal partition piecewise constant piecewise continuous functions polynomials projection theorem projector prove quadratic form relation Remark respect s-axis Section self-adjoint operator sequence of vectors space of functions spectral representation spectrum statement strict adjoint strictly self-adjoint subspace Suppose theory tion transformation triangle inequality unit form values vanishes variable zero