## Linear Operators: General theory |

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Page 419

Then the l '

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

subset of r , and > 0 . The terms l - 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 , ε ) = { 9 } \ / ( p ) - | ( 9 ) / < € , 1 € A } , where pe X , A is a finite

subset of r , and > 0 . The terms l - open and I - closed subsets of X , I - continuous

...

Page 420

in the X *

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 , ( x ) = x ( y ) , 2 € , and the

subspace I ...

in the X *

**topology**if and only if lim , xa = 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 , ( x ) = x ( y ) , 2 € , and the

subspace I ...

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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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