## Linear Operators: General theory |

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Page 224

It will be assumed that the reader is familiar with the elementary theory of

complex valued

on G and with values in X is said to be

partial ...

It will be assumed that the reader is familiar with the elementary theory of

complex valued

**analytic**functions of one complex variable ... A function f definedon G and with values in X is said to be

**analytic**on G if f is continuous and the firstpartial ...

Page 228

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

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

partial derivatives of in 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 in

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

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

partial ...

Page 230

Nelson Dunford. 20 of n The largest number n such that a_ini # 0 is called the

order of the pole 20 If no a , with p < 0 is non - zero , and if we put | ( 20 ) = do ,

then f becomes

removable .

Nelson Dunford. 20 of n The largest number n such that a_ini # 0 is called the

order of the pole 20 If no a , with p < 0 is non - zero , and if we put | ( 20 ) = do ,

then f becomes

**analytic**in 12—201 < r , so that the singularity at z = 2 , isremovable .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero