## Linear Operators: General theory |

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Page 72

16 Let X be a normed linear space which is not assumed to be complete , and let

3 be a closed

also . 17 DEFINITION . If X is a normed linear space , and Z ÇX , the set Zt = { x * x

...

16 Let X be a normed linear space which is not assumed to be complete , and let

3 be a closed

**subspace**of X . Then , if Z and X / 3 are complete , show that X isalso . 17 DEFINITION . If X is a normed linear space , and Z ÇX , the set Zt = { x * x

...

Page 436

6 Let X be a B - space , and X , a

is the same as the relative X * topology of X . 7 Let X be a linear space , and I a

total

...

6 Let X be a B - space , and X , a

**subspace**of X . Show that the X * topology of X ,is the same as the relative X * topology of X . 7 Let X be a linear space , and I a

total

**subspace**of X * . Show that a set A CX is I - bounded if and only if | ( A ) is a...

Page 513

16 If Y is a closed

N is finite dimensional , it does not follow that Y is closed . 17 Let X be a B - space

and ...

16 If Y is a closed

**subspace**of a B - space and N is a finite dimensional**subspace**, then Y O N is a closed**subspace**. If Y O N is a closed**subspace**, andN is finite dimensional , it does not follow that Y is closed . 17 Let X be a B - space

and ...

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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