The current booklet relies on lectures given by way of the writer on the collage of Tokyo prior to now ten years. it's meant as a textbook to be studied by way of scholars all alone or for use in a direction on practical research, i. e. , the overall idea of linear operators in functionality areas including salient positive aspects of its program to different fields of recent and classical research. precious necessities for the interpreting of this publication are summarized, without or with facts, in bankruptcy zero less than titles: Set thought, Topo logical areas, degree areas and Linear areas. Then, beginning with the bankruptcy on Semi-norms, a normal idea of Banach and Hilbert areas is gifted in reference to the idea of generalized services of S. L. SOBOLEV and L. SCHWARTZ. whereas the ebook is basically addressed to graduate scholars, it's was hoping it will probably end up valuable to investigate mathe maticians, either natural and utilized. The reader might move, e. g. , from bankruptcy IX (Analytical conception of Semi-groups) on to bankruptcy XIII (Ergodic idea and Diffusion thought) and to bankruptcy XIV (Integration of the Equation of Evolution). Such fabrics as "Weak Topologies and Duality in in the community Convex areas" and "Nuclear areas" are awarded within the type of the appendices to bankruptcy V and bankruptcy X, respectively. those should be skipped for the 1st examining through those people who are quite within the program of linear operators.