## Linear Operators: General theory |

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Page 372

769 )

an Euclidean space . This , in particular , implies the completeness . Theorem 3 .

5 , which characterizes locally compact B - spaces is due to F . Riesz [ 4 ] .

769 )

**proved**that any finite dimensional topological linear space is equivalent toan Euclidean space . This , in particular , implies the completeness . Theorem 3 .

5 , which characterizes locally compact B - spaces is due to F . Riesz [ 4 ] .

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This generalizes and abstracts a result

0 , 1 ] by E . Fischer [ 2 ] . The fact that a linear manifold which is not dense in the

entire space has a non - zero orthogonal complement (

This generalizes and abstracts a result

**proved**for closed linear manifolds in L2 [0 , 1 ] by E . Fischer [ 2 ] . The fact that a linear manifold which is not dense in the

entire space has a non - zero orthogonal complement (

**proved**in 4 . 4 ) was ...Page 385

27 . They are essentially due , at least in the real case , to Stone [ 1 ] , although

his terminology and proofs often differ from that given here . It should be

mentioned that Theorem 6 . 22 was

slightly later .

27 . They are essentially due , at least in the real case , to Stone [ 1 ] , although

his terminology and proofs often differ from that given here . It should be

mentioned that Theorem 6 . 22 was

**proved**independently by Cech [ 1 ] onlyslightly later .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero