## Linear Operators: General theory |

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Page 17

A topological space X is said to be locally

neighborhood whose closure is

intersection property 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 property 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 x ( TS) is

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

Page 485

( Gantmacher ) An operator in B ( X , Y ) is weakly

is weakly

sphere S * of Y * is Y -

.7 ...

( 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 ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element 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 o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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