This book was written for students of mathematics and physics who have a basic knowledge of analysis and linear algebra. It can be used as a textbook for courses and/or seminars in functional analysis. Starting from metric spaces, it proceeds quickly to the central results of the field, including the theorem of Hahn-Banach. The spaces (p p ( ,(), ( )' and Sebolov spaces are introduced. A chapter on spectral theory contains the Riesz theory of compact operators, basic facts on Banach and C-algebras, and the spectral representation for bounded normal and unbounded self-adjoint operators for Hilbert spaces. A discussion of locally convex spaces and their duality theory provides the basis for a comprehensive treatment of Frechet spaces and their duals.