## Linear Operators: General theory |

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Page 54

Since the

proof is complete . Q . E . D . We ... A linear

is continuous if and only if it

Since the

**mapping**x + x / k , is a homeomorphism of X with itself , by 10 ( ii ) , theproof is complete . Q . E . D . We ... A linear

**mapping**of one F - space into anotheris continuous if and only if it

**maps**bounded sets into bounded sets . PROOF .Page 55

x in X . Then lim x0 Tnx = 0 uniformly for n = 1 , 2 , . . . , and T is a continuous

linear

the linearity of the operators Tn . For each X , the sequence { Tmx } is convergent

...

x in X . Then lim x0 Tnx = 0 uniformly for n = 1 , 2 , . . . , and T is a continuous

linear

**map**of X into y . Proof . The linearity of T is an immediate consequence ofthe linearity of the operators Tn . For each X , the sequence { Tmx } is convergent

...

Page 478

The

Proof . The linear functional y * T is continuous ( 1 . 4 . 17 ) , and hence T * y * £ X

* . The

The

**mapping**T →T * is an isometric isomorphism of B ( X , Y ) into B ( Y * , X * ) .Proof . The linear functional y * T is continuous ( 1 . 4 . 17 ) , and hence T * y * £ X

* . The

**map**T →T * is clearly linear . From II . 3 . 15 , Tx = sup \ y * Tx , and thus ...### 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 |

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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