## Linear Operators, Part 1 |

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Page 17

A topological space X is said to be locally

neighborhood whose closure is

intersection properly if every finite subfamily has a non - void intersection . A

subset of X is called ...

A topological space X is said to be locally

**compact**if every point has aneighborhood whose closure is

**compact**. A family of sets has the finiteintersection properly if every finite subfamily has a non - void intersection . A

subset of X is called ...

Page 483

If T is weakly

) is

If T is weakly

**compact**, then TS is**compact**in the Y * topology of Y and thus z ( TS) is

**compact**and hence closed in the Y * topology of Y ** . Thus if T is weakly**compact**, ( i ) yields 7 ** ( S , ) Ç ( TS ) . According to Theorem V.4.5 , S , = S ** ...Page 485

( Gantmacher ) An operator in B ( x , y ) is weakly

is weakly

sphere S * of Y * is y -

7 that ...

( Gantmacher ) An operator in B ( x , y ) is weakly

**compact**if and only if its adjointis weakly

**compact**. Proof . Let T be weakly**compact**. Since the closed unitsphere S * of Y * is y -

**compact**( V.4.2 ) , it follows from Lemma 7 and Lemma 1.5.7 that ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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