## Linear Operators: General theory |

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Page 227

The Cauchy integral

integral theorem just as in the case of complex valued analytic functions . Let Uį ,

. . . Un be a finite family of bounded open sets , with each U ; having boundary B ,

as ...

The Cauchy integral

**formula**may also be proved directly from the Cauchyintegral theorem just as in the case of complex valued analytic functions . Let Uį ,

. . . Un be a finite family of bounded open sets , with each U ; having boundary B ,

as ...

Page 228

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

Cauchy integral

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

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

Cauchy integral

**formula**in several variables , the following convergence theoremof Weierstrass : Let In be a uniformly bounded sequence of vector valued ...

Page 407

There exists a corresponding Wiener - integral

more general parabolic initial value problem . OM 1 22 [ * ] F ( x , t ) = = = F ( x , t )

+ V ( x , t ) F ( x , t ) , 20 ; 4 Əx2 " F ( x , 0 ) = f ( x ) , V being a given coefficient ...

There exists a corresponding Wiener - integral

**formula**for the solution F of themore general parabolic initial value problem . OM 1 22 [ * ] F ( x , t ) = = = F ( x , t )

+ V ( x , t ) F ( x , t ) , 20 ; 4 Əx2 " F ( x , 0 ) = f ( x ) , V being a given coefficient ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero