## Linear Operators: General theory |

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Page 24

Nelson Dunford, Jacob T. Schwartz. PROôF . If x € X , there is a sequence of

points an € A with an + x . Since { am } is a Cauchy sequence , and f is uniformly

complete ...

Nelson Dunford, Jacob T. Schwartz. PROôF . If x € X , there is a sequence of

points an € A with an + x . Since { am } is a Cauchy sequence , and f is uniformly

**continuous**, the sequence { f ( an ) } is also a Cauchy sequence . Since Y iscomplete ...

Page 58

The map prx : [ w , Tx ] → x of G onto X is one - to - one , linear , and

.8.3 ) . Hence , by Theorem 2 , its inverse prą ' is

The map prx : [ w , Tx ] → x of G onto X is one - to - one , linear , and

**continuous**( I.8.3 ) . Hence , by Theorem 2 , its inverse prą ' is

**continuous**. Thus T = pry prz ' is**continuous**( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space ...Page 157

The space E ( u ) is therefore a complete metric space . If 2 is an additive vector

or scalar valued function on which is u -

E A F ) ...

The space E ( u ) is therefore a complete metric space . If 2 is an additive vector

or scalar valued function on which is u -

**continuous**, then à is defined and**continuous**on the metric space ( u ) . To see this note first that the identities v ( u ,E A F ) ...

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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