1.Topological vector spaces. 2.Completeness. 3.Convexity. 4.Duality in Banach spaces. 5.Some applications. 6.Test functions and distributions. 7.Fourier transforms. 8.Applications to differential equations. 9.Tauberian theory. 10.Banach algebras. 11.Commutative Banach algebras. 12.Bounded operators on a Hilbert Spaces. 13.Unbounded operators.