## Linear Operators, Part 1 |

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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 478

The adjoint T * of a linear operator T in B ( X , Y ) is the

defined by T * y * = y * T . 2 LEMMA . The

isomorphism of B ( X , Y ) into B ( Y * , X * ) . Proof . The linear functional y * T is

continuous ...

The adjoint T * of a linear operator T in B ( X , Y ) is the

**mapping**from Y * to X *defined by T * y * = y * T . 2 LEMMA . The

**mapping**T ^ T * is an isometricisomorphism of B ( X , Y ) into B ( Y * , X * ) . Proof . The linear functional y * T is

continuous ...

Page 493

79 Conversely , if ( a ) and ( b ) are satisfied for the

u ( • ) æ * , then it follows that for each fixed f e C ( S ) the

u ( ds ) a * is continuous in the X topology of X * and therefore ( V . 3 .

79 Conversely , if ( a ) and ( b ) are satisfied for the

**mapping**which sends x * intou ( • ) æ * , then it follows that for each fixed f e C ( S ) the

**mapping*** * ▻S $ t ( s )u ( ds ) a * is continuous in the X topology of X * and therefore ( V . 3 .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### 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

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