## History of Functional AnalysisHistory of Functional Analysis presents functional analysis as a rather complex blend of algebra and topology, with its evolution influenced by the development of these two branches of mathematics. The book adopts a narrower definition—one that is assumed to satisfy various algebraic and topological conditions. A moment of reflections shows that this already covers a large part of modern analysis, in particular, the theory of partial differential equations. This volume comprises nine chapters, the first of which focuses on linear differential equations and the Sturm-Liouville problem. The succeeding chapters go on to discuss the ""crypto-integral"" equations, including the Dirichlet principle and the Beer-Neumann method; the equation of vibrating membranes, including the contributions of Poincare and H.A. Schwarz's 1885 paper; and the idea of infinite dimension. Other chapters cover the crucial years and the definition of Hilbert space, including Fredholm's discovery and the contributions of Hilbert; duality and the definition of normed spaces, including the Hahn-Banach theorem and the method of the gliding hump and Baire category; spectral theory after 1900, including the theories and works of F. Riesz, Hilbert, von Neumann, Weyl, and Carleman; locally convex spaces and the theory of distributions; and applications of functional analysis to differential and partial differential equations. This book will be of interest to practitioners in the fields of mathematics and statistics. |

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### Contents

1 | |

9 | |

22 | |

THE EQUATION OF VIBRATING MEMBRANES | 47 |

THE IDEA OF INFINITE DIMENSION | 71 |

THE CRUCIAL YEARS AND THE DEFINITION OF HILBERT SPACE | 97 |

DUALITY AND THE DEFINITION OF NORMED SPACES | 121 |

SPECTRAL THEORY AFTER 1900 | 144 |

LOCALLY CONVEX SPACES AND THE THEORY OF DISTRIBUTIONS | 210 |

APPLICATIONS OF FUNCTIONAL ANALYSIS TO DIFFERENTIAL AND PARTIAL DIFFERENTIAL EQUATIONS | 233 |

REFERENCES | 280 |

299 | |

306 | |

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apply arbitrary Banach algebra Banach spaces bilinear bilinear form boundary condition bounded called Carleman Cauchy chap closed coefficients commutative compact support complete concept considered constant continuous function convergence corresponding defined definition derivatives differential operator Dirichlet problem distribution domain eigenfunctions eigenvalues elliptic existence finite dimensional finite number formula Fourier transform Fréchet Fredholm func function f Functional Analysis Gelfand geometric given Green function Helly hermitian operator Hilbert space holomorphic idea inequality infinite integral equations interval kernel later limit linear equations linear form linear mapping Math mathematicians matrix method Neumann normal normed space notion obtained orthogonal paper partial differential equations Poincaré proof properties proved pseudo-differential operator real numbers Riesz satisfying scalar Schwarz's second order self-adjoint sequence shows solution spectral theory spectrum subset tends theorem tion uniform convergence unique values variables vector space vector subspace vol.II weak topology Weyl writes