## Linear Operators: General theory |

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Page 419

Then the I

form N ( p ; A , ε ) = { 9 | | ( p ) - 1 ( 0 ) 1 < E , JE A } , where p e X , A is a finite

subset of In and a > 0 . The terms l - open and I - closed subsets of X , C -

continuous ...

Then the I

**topology**of X is the**topology**obtained by taking as base all sets of theform N ( p ; A , ε ) = { 9 | | ( p ) - 1 ( 0 ) 1 < E , JE A } , where p e X , A is a finite

subset of In and a > 0 . The terms l - open and I - closed subsets of X , C -

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 yt , then each element y e Y determines the linear functional f , on X defined by

...

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

**topology**if and only if limą XQ = xweakly in the sense of Definition II . 3 . 25 . On the other hand , if X is a subspace

of yt , then each element y e Y determines the linear functional f , on X defined by

...

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 | |

31 other sections not shown

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### Common terms and phrases

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