1.Generalities on topological vector spaces. 2.Norms and semi-norms. 3.Locally convex spaces. 4.Topological algebras and multipliocatively locally convex algebras. 5.Some examples. 6.Regularity of topological vector-spaces. 7.Quotient topological vector spaces. 8.Cartesian products and direct sums of topological vector-spaces. 9.Continuity of linear mapping. 10.The classical analytic form of the Hahn-Banach theorem for vector –spaces. 11.The classical analytic form of the Hahn-Banach theorem for semi-normed spaces. 12.The analytic form of the Hahn-Banach theorem for locally convex spaces. 13.The Hahn-Banach theorem as an approximation theorem for finite linear combinations. 14.The geometrical form of the Hahn-Banach theorem for open convex sets. 15.The geometrical form of the of the Hahn-Banach theorem for closed convex-sets. 16.The Hahn-Banach theorem as an approximation theorem for finite convex or positive linear combinations. 17.Remark about the need of local convexity. 18.Weak topologies. 19.Duality between vector spaces. 20.Duality and polar subsets. 21.Duality and orthogonal vector-spaces. 22.The bourbaki-Alaoglu theorem. 23.Metrizability of locally convex spaces and frechet spaces. 24.Complete topological-vector spaces. 25.Bounded sets. 26.The Banach-Steinhaus theorem. 27.Barreled spaces and Banach-Steinhaus property. 28.Barreled spaces of continuous functions. 29.Bornological spaces and the continuity of bounded linear mapping. 30.Bornological spaces of continuous functions. 31.Duality and reflexive spaces. 32.Duality and reflexive spaces. 33.The Bolzano-Weierstrass property and Montel-Spaces. 34.Finite-dimensional topological vector spaces. 35.Quasi-complete topological vector-spaces. 36.Projective or inverse limits of locally convex spaces. 37.Inductive or direct limits of locally convex spaces. 38.Some fundamental examples of inductive limits. 39.Some properties of inductive limits. 40.A summary of the main classes of locally-convex-spaces.