## Linear Operators, Part 1 |

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Page 419

Then the l '

form N ( p ; A , € ) = { 9 } \ / ( p ) -1 ( 9 ) < € , 1 € A } , where p e X , A is a finite

subset of T ' , and > 0 . The terms I - open and I - closed subsets of X , I -

continuous ...

Then the l '

**topology**of X is the**topology**obtained by taking as base all sets of theform N ( p ; A , € ) = { 9 } \ / ( p ) -1 ( 9 ) < € , 1 € A } , where p e X , A is a finite

subset of T ' , and > 0 . The terms I - open and I - closed subsets of X , I -

continuous ...

Page 420

Nelson Dunford, Jacob T. Schwartz. in the X *

weakly in the sense of Definition II.3.25 . On the other hand , if X is a subspace of

y * , then each element ye Y determines the linear functional f , on X defined by 1

...

Nelson Dunford, Jacob T. Schwartz. in the X *

**topology**if and only if lim , Xa = xweakly in the sense of Definition II.3.25 . On the other hand , if X is a subspace of

y * , then each element ye Y determines the linear functional f , on X defined by 1

...

Page 512

T → is compact in the strong operator

sequentially compact in the weak operator

the weak operator

X , Y ) ...

T → is compact in the strong operator

**topology**. If Y is also separable , A issequentially compact in the weak operator

**topology**if and only if A is compact inthe weak operator

**topology**. 6 If Y is reflexive , then the closed unit sphere of B (X , Y ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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

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 isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space 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