## Functional AnalysisThe present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i. e. , the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topo logical Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathe maticians, both pure and applied. The reader may pass, e. g. , from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators. |

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### Contents

Preliminaries | 1 |

Applications of the BaireHausdorff Theorem | 68 |

The Orthogonal Projection and F Riesz Representation Theo | 81 |

Copyright | |

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apply assume B-space Banach space belongs bounded bounded linear operator called Chapter clear closed compact complex condition consider constant contains continuous linear converges Corollary defined Definition denote dense determined differential distribution domain element equal equation equi-continuous ergodic Example exists extension fact finite fixed given gives hand Hence Hilbert space ideal implies inequality infinitesimal integral inverse Lemma linear functional linear operator linear space mapping Math maximal mean measure Moreover neighbourhood norm obtain open set positive preceding Proof Proposition proved range Remark representation resolvent respect s-lim satisfies self-adjoint semi-group semi-norm sequence solution strong strongly subset sufficiently Suppose symmetric Theorem theory topological topological space transform uniformly uniquely unit vector YOSIDA