## Linear Operators: General theory |

### From inside the book

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Page 224

Functions of a Complex Variable In some of the chapters to follow, and especially

in Chapter VII, we shall use extensions of certain well-known results in the theory

of

Functions of a Complex Variable In some of the chapters to follow, and especially

in Chapter VII, we shall use extensions of certain well-known results in the theory

of

**analytic**functions of a complex variable to the case where the functions are ...Page 228

Let V be a bounded open subset of U whose closure is contained in U. If f„

converges at each point of V to a function f on U then f is

partial derivatives of /„ of arbitrarily large order converge to the corresponding

partial ...

Let V be a bounded open subset of U whose closure is contained in U. If f„

converges at each point of V to a function f on U then f is

**analytic**in U, and thepartial derivatives of /„ of arbitrarily large order converge to the corresponding

partial ...

Page 588

Consequently, it is possible to solve for the tif, and obtain in each case the

quotient of an

reciprocal, and therefore each ttj, is

Consequently, it is possible to solve for the tif, and obtain in each case the

quotient of an

**analytic**function by A (fx). Since, for | < 62, the function A (ft ) ^ 0, itsreciprocal, and therefore each ttj, is

**analytic**. Thus the tiS are**analytic**for \ft\ < d2.### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero