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The scalar product inner product
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adjoint operator arbitrary bounded linear operator bounded operator bounded variation Cayley transform closure commute completely continuous operator construct corresponding countable set defined everywhere definition denote dense in H distribution function domain eD(A eD(T eigenvalue eigenvectors element g equality equation everywhere in H finite interval finite-dimensional formula function a(t Hence it follows Hilbert space holds identity implies inequality invariant sub-space isometric operator J_oo Lemma linear envelope linear functional linear manifold linearly independent mapping matrix measure zero relative multiplicity natural number non-decreasing function non-zero eigenvalue norm obtain operator defined operator in H operator Q ortho-projectors orthogonal orthonormal basis orthonormal system polynomials projection operator Proof prove satisfies the conditions scalar product self-adjoint operator separable space sequence of vectors simple spectrum space H spectral type sub-space G suppose symmetric operator uniquely unitary operator vector vector h e H vector-function whole space