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IntroduchonChapter One Operator theory in finite��dimensional vector spaces��1��Vector spaces and normed vector spaces1��Basic nohons2��Bases3��Linear manifolds4��Convergence and norms5��Topological nohons in a normed space6��Infinite series of vectors7��Vector��valued funchons��2��Linear forms and the adjoint space1��Linear forms2��The adjoint space3��The adjoint basis4��The adjoint space of a normed space5��The convexity of balls6��The second adjoint space��3��Linear operators1��Definihons��Matrix representations2��Linear operations on operators3��The algebra of linear operators4��Projections��Nilpotents5��Invariance��Decomposihon6��The adjoint operator��4��Analysis with operators1��Convergence and norms for operators2��The norm of T3��Examples of norms4��Infinte series of operators5��Operator��valued functions6��Pairs of projechons��5��The eigenvalue problem1��Definihons2��The resolvent3��Singularities of the resolvent4��The canorucal form of an operator5��The adjoint problem6��Functions of an operator7��Similarity transformations��6��Operators in unitary spaces1��Unitary spaces2��The adjoint space3��Orthonormal families4��Linear operators5��Symmetric forms and symmetric operators6��Unitary��isometric and normal operators7��Projections8��Pairs of projections9��The eigenvalue problem10��The minimax principleChapter Two Perturbatlon theory in a finite��dimensional space��1��Analyhc perturbahon of eigenvalues1��The problem2��Singularities of the eigenvalues3��Perturbation of the resolvent4��Perturbation of the eigenprojections5��Singularities of the eigertprojections6��Remarks and examples7��The case of T��x�� linear in x8��Summary��2��Perturbation series1��The total projectin for the A��group2��The weighted mean of eigenvalues3��The reduction process4��Formulas for higher approximahons5��A theorem of MOTZKIN��TAUSSKY6��The ranks of the coefficients of the perturbation series��3��Convergence radu and error estimates1��Simple estimates2��The method of majorizing series3��Estimates on eigenvectors4��Further error eshmates5��The special case of a normal unperturbed operator6��The enumerahve method��4��Similarity transformations of the eigenspaces and eigenvectors1��Eigenvectors2��Transformation funchons3��Soluhon of the dffierential equahon4��The transformation function and the reduction process5��Simultaneous transformahon for several projections6��Diagonalization of a holomorphic matrix function��5��Non��analytic perturbations1��Continuity of the eigenvalues and the total projechon2��The numbering of the eigenvalues3��Conhnuity of the eigenspaces and eigenvectors4��Differentiability at a point5��Differenhability in an interval���� Chapter Three Introduction to the theory of operators in Banach spacesChapter Four Stability theoremsChapter Five Operators in Hilbert spacesChapter Six Sesquilinear forms in Hilbert spaces and associated operatorsChapter Seven Analytic perturbation theoryChapter Eight Asymptotic perturbation theoryChapter Nine Perturbation theory for semigroups of operatorsChapter Ten Perturbauuon of continuous spectra and unitary equivalenceSupplementary Notes