## Linear Operators, Part 1 |

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Page 195

It follows that if f is a function defined on Euclidean n -

integrable with respect to n -

integral Set ( s ) 2 ( ds ) is equal to the “ iterated ” integral St . { . . . { St . 07 ( 81 , . .

.

It follows that if f is a function defined on Euclidean n -

**dimensional**space E andintegrable with respect to n -

**dimensional**Lebesgue measure 2 , the “ multiple ”integral Set ( s ) 2 ( ds ) is equal to the “ iterated ” integral St . { . . . { St . 07 ( 81 , . .

.

Page 245

An n -

operator on a finite

bı , . . . bn } be a Hamel basis for the finite

...

An n -

**dimensional**B - space is equivalent to En . 4 COROLLARY . Every linearoperator on a finite

**dimensional**normed linear space is continuous . Proof . Let {bı , . . . bn } be a Hamel basis for the finite

**dimensional**normed linear space X so...

Page 246

Then the

X * * ( II . 3 . 19 ) , the

spaces with zero

Then the

**dimension**of X * * is finite , and , since X is equivalent to a subspace ofX * * ( II . 3 . 19 ) , the

**dimension**of X is finite . ... 11 ) of infinite**dimensional**F -spaces with zero

**dimensional**conjugate spaces . The following corollary was ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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