## Linear Operators: General theory |

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Page 477

strong , and the weak

uniform

the strong

...

strong , and the weak

**operator topologies**in B ( X , Y ) . It is evident that theuniform

**operator topology**is stronger than the strong**operator topology**, and thatthe strong

**operator topology**is stronger than the weak**operator topology**. With its...

Page 511

Exercises 1 The space B ( X , Y ) is algebraically isomorphic to a subspace of the

product Prex Ye , where Yæ = Y , under the mapping T → Prex Tx . Show that the

strong

Exercises 1 The space B ( X , Y ) is algebraically isomorphic to a subspace of the

product Prex Ye , where Yæ = Y , under the mapping T → Prex Tx . Show that the

strong

**topology**of**operators**in B ( X , Y ) is identical with the usual product ...Page 512

is compact in the strong

sequentially compact in the weak

the weak

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

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero