## Linear Operators, Part 1 |

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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 ...

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 thefunctions ...

Page 230

If no 20 is The largest number n such that a_inl # 0 is called the order of the pole

ар with p < 0 is non - zero , and if we put f ( 20 ) = do , then f becomes

- zo ! < r , so that the singularity at z = removable . If an o for p = 0 , 2 , is called a ...

If no 20 is The largest number n such that a_inl # 0 is called the order of the pole

ар with p < 0 is non - zero , and if we put f ( 20 ) = do , then f becomes

**analytic**in z- zo ! < r , so that the singularity at z = removable . If an o for p = 0 , 2 , is called a ...

Page 586

Nelson Dunford, Jacob T. Schwartz. Then there exists a 8 > 0 such that if \ ul < d ,

then Ū CO ( T ( u ) ) and R ( 2 ; T ( u ) ) is an

PROOF . By Lemma 3 , there is a & such that if sul < 81 , then U CO ( T ( u ) ) .

Nelson Dunford, Jacob T. Schwartz. Then there exists a 8 > 0 such that if \ ul < d ,

then Ū CO ( T ( u ) ) and R ( 2 ; T ( u ) ) is an

**analytic**function of u for each 1 € U.PROOF . By Lemma 3 , there is a & such that if sul < 81 , then U CO ( T ( u ) ) .

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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