Preliminaries.- Semi-norms.- Applications of the Baire-Hausdorff Theorem.- The Orthogonal Projection and F. Riesz' Representation Theorem.- The Hahn-Banach-Theorems.-Strong Convergence and Weak Convergence.- Fourier Transform and Differential Equations.- Dual Operators.- Resolvent and Spectrum.- Analytical Theory of Semi-groups.- Compact Operators.- Normed Rings and Spectral Representation.- Other Representation Theorems in Linear Spaces.- Ergodic Theory and Diffusion Theory.- The Integration of the Equations of Evolution.
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B-space Baire bounded linear operator bounded set closed linear operator closed linear subspace compact complex numbers contains continuous function continuous linear functional continuous semi-norm converges convex linear topological Corollary defined Definition denote dense differential operator domain eigenvalue element ergodic theorem finite fo(x Fourier transform functional f Hence Hilbert space holomorphic implies inequality infinitesimal integral inverse Lemma linear subspace linear topological space locally convex linear mapping Math matrix maximal ideal neighbourhood non-negative normed linear space O-lim obtain open set operator H positive constant positive number Proof Proposition real number real-valued representation s-lim satisfies the condition self-adjoint operator semi-group of class semi-norm ſ║ solution strongly subset symmetric operator theorem in Chapter topological space T║x uniformly uniquely determined unitary weak topology weakly YosidA