## Linear Operators: General theory |

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Page 26

Hence the

. Convergence and Uniform Convergence of Generalized Sequences The notion

of convergence introduced in Definition 6 . 5 is not sufficiently general for all our ...

Hence the

**identity**is a homeomorphism and the space is metrizable . Q . E . D . 7. Convergence and Uniform Convergence of Generalized Sequences The notion

of convergence introduced in Definition 6 . 5 is not sufficiently general for all our ...

Page 249

Nelson Dunford, Jacob T. Schwartz. Proof . The

+ 2 \ y / 2 , X , Y EH , called the parallelogram

the axioms . If 8 = inf | x - k | the preceding

Nelson Dunford, Jacob T. Schwartz. Proof . The

**identity**1x + ya + \ x - y2 = 2 | 212+ 2 \ y / 2 , X , Y EH , called the parallelogram

**identity**, follows immediately fromthe axioms . If 8 = inf | x - k | the preceding

**identity**shows that keK | ki - k ; \ 2 = 2 ...Page 479

... and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X * .

When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - l exists and is in B (

Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

... and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X * .

When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - l exists and is in B (

Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

**identity**in Y * and ( T ...### What people are saying - Write a review

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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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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