## 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 ) C x ( TS ) . According to Theorem V . 4 . 5 , S , = $ ** ...

Page 494

From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally

weakly

By Theorem 4.8 this implies that T is a weakly

From IV.10.2 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 .By Theorem 4.8 this implies that T is a weakly

**compact**operator . Q.E.D. given ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

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### 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero