## Linear Operators: General theory |

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Page 171

Nelson Dunford, Jacob T. Schwartz. 23

is a o - field , and that u is countably additive and bounded .

continuous function is u - measurable . Show that £ * contains every Borel set .

Nelson Dunford, Jacob T. Schwartz. 23

**Suppose**that S is a metric space , that Eis a o - field , and that u is countably additive and bounded .

**Suppose**that everycontinuous function is u - measurable . Show that £ * contains every Borel set .

Page 360

21 Show that if | Só En ( x , x ) dz 5 M , then the convergence of Snf for a given c .

o . n . system is localized if and only if max En ( x , y ) SM < oo for each ε > 0 . la -

v12€ 22

21 Show that if | Só En ( x , x ) dz 5 M , then the convergence of Snf for a given c .

o . n . system is localized if and only if max En ( x , y ) SM < oo for each ε > 0 . la -

v12€ 22

**Suppose**that ( S . J ) ( x ) + f ( x ) uniformly for every f in AC . Show that ...Page 598

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

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

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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