## Linear Operators: General theory |

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Page 169

8 Show that S can be the union of an increasing sequence of null sets even if u is

not identically

subset of X , and if f - 1 ( G ) is in E for each open subset G of X , then f is totally ...

8 Show that S can be the union of an increasing sequence of null sets even if u is

not identically

**zero**. 9 Show that if f is defined on S and has values in a compactsubset of X , and if f - 1 ( G ) is in E for each open subset G of X , then f is totally ...

Page 204

Since u ( F . n ) = Ss , In ( San Jua , ( dsm , ) does not converge to

0 sin ( sq ) 1 it follows from the dominated convergence theorem ( 6 . 16 ) that

there is a point som in S for which fn ( $ 0 , ) is defined for all n and for which the ...

Since u ( F . n ) = Ss , In ( San Jua , ( dsm , ) does not converge to

**zero**and since0 sin ( sq ) 1 it follows from the dominated convergence theorem ( 6 . 16 ) that

there is a point som in S for which fn ( $ 0 , ) is defined for all n and for which the ...

Page 452

... a nonzero continuous linear functional tangent to K at p . If A is a subset of X ,

and p is in A , then there exists a non -

to A at p if and only if the cone B with vertex p generated by A is not dense in X . !

... a nonzero continuous linear functional tangent to K at p . If A is a subset of X ,

and p is in A , then there exists a non -

**zero**continuous linear functional tangentto A at p if and only if the cone B with vertex p generated by A is not dense in X . !

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 other sections not shown

### 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

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