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 Equation of Evolution.