## Linear Operators: General theory |

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Page 419

Nelson Dunford, Jacob T. Schwartz, William G. Bade. 2 DEFINITION . Let X be a

linear vector space , and let T be a total subspace of X * . Then the l '

X is the

...

Nelson Dunford, Jacob T. Schwartz, William G. Bade. 2 DEFINITION . Let X be a

linear vector space , and let T be a total subspace of X * . Then the l '

**topology**ofX is the

**topology**obtained by taking as base all sets of the form N ( p ; A , ε ) = { q...

Page 420

Nelson Dunford, Jacob T. Schwartz, William G. Bade. in the X *

only if lim , xe = 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 ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade. in the X *

**topology**if andonly if lim , xe = 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 ...

Page 512

is compact in the strong operator

sequentially compact in the weak operator

the weak operator

X , Y ) is ...

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 ) is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero