## Linear Operators: General theory |

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Page 477

strong, and the weak

strong

strong, and the weak

**operator topologies**in B(S, *?)). It is evident that the uniform**operator topology**is stronger than the strong**operator topology**, and that thestrong

**operator topology**is stronger than the weak**operator topology**. With its ...Page 511

Exercises 1 The space ?)) is algebraically isomorphic to a subspacc of the

product where |)B = under the mapping T PMx Tx. Show that the strong

of

strong ...

Exercises 1 The space ?)) is algebraically isomorphic to a subspacc of the

product where |)B = under the mapping T PMx Tx. Show that the strong

**topology**of

**operators**in B(K, "2)) is identical with the usual product**topology**where thestrong ...

Page 512

is compact in the strong

sequentially compact in the weak

the weak

3E, 3J) is ...

is compact in the strong

**operator topology**. If 3J is also separable, A issequentially compact in the weak

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

**operator topology**. 6 If 3) is reflexive, then the closed unit sphere of B(3E, 3J) is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact