## 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 485

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

is weakly

sphere S * of Y * is Y -

.

( 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.

Page 494

2 we conclude that T * maps the unit sphere of X * into a conditionally weakly

Theorem 4 . 8 this implies that T is a weakly

THEOREM .

2 we conclude that T * maps the unit sphere of X * into a conditionally weakly

**compact**set of rca ( S ) , and therefore T * is a weakly**compact**operator . ByTheorem 4 . 8 this implies that T is a weakly

**compact**operator . Q . E . D . 4THEOREM .

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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