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The scalar product inner product
90 other sections not shown
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