## Linear Operators: General theory |

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Page 227

The same use of functionals proves the Cauchy integral

588- % where z is a point of the bounded open set U whose boundary B consists

of a finite number of closed rectifiable Jordan curves oriented in the positive ...

The same use of functionals proves the Cauchy integral

**formula**: 1 fa ) f ( x ) = -588- % where z is a point of the bounded open set U whose boundary B consists

of a finite number of closed rectifiable Jordan curves oriented in the positive ...

Page 228

Un . Just as in the classical one variable case we can prove , using this Cauchy

integral

Weierstrass : Let In be a uniformly bounded sequence of vector valued functions

each ...

Un . Just as in the classical one variable case we can prove , using this Cauchy

integral

**formula**in several variables , the following convergence theorem ofWeierstrass : Let In be a uniformly bounded sequence of vector valued functions

each ...

Page 407

There exists a corresponding Wiener - integral

more general parabolic initial value problem . a 1 22 F ( x , t ) F ( x , t ) + V ( .x , t )

F ( x , t ) , t 20 ; at 4 дr2 F ( x , 0 ) = f ( r ) , V being a given coefficient function .

There exists a corresponding Wiener - integral

**formula**for the solution F of themore general parabolic initial value problem . a 1 22 F ( x , t ) F ( x , t ) + V ( .x , t )

F ( x , t ) , t 20 ; at 4 дr2 F ( x , 0 ) = f ( r ) , V being a given coefficient function .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions 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 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