## Linear Operators: General theory |

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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 Ā is compact inthe weak

**operator topology**. 6 If Y is reflexive , then the closed unit sphere of B (X , Y ) is ...

Page 513

topology , but that { A A * } does not converge to zero in this topology even though

{ A * An } converges to zero in the strong

space , the mapping T → T * of B ( H ) into itself is continuous with either the ...

topology , but that { A A * } does not converge to zero in this topology even though

{ A * An } converges to zero in the strong

**operator topology**. 12 If H is a Hilbertspace , the mapping T → T * of B ( H ) into itself is continuous with either the ...

Page 598

Suppose that gn ( T ) converges in the uniform

operator . Let à € o ( T ) , and lim - & n ( a ) + 0 . Show that à is a pole of o ( T ) ,

and that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII . 5 .

35 . ) ...

Suppose that gn ( T ) converges in the uniform

**operator topology**to a compactoperator . Let à € o ( T ) , and lim - & n ( a ) + 0 . Show that à is a pole of o ( T ) ,

and that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII . 5 .

35 . ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero