## Linear Operators, Part 1 |

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Page 373

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 .

Page 608

In the case of a bounded or unbounded normal operator in a Hilbert space ,

many of the results of this section become simpler and can be

directly by other methods . With these hypotheses considerable extension is

possible .

In the case of a bounded or unbounded normal operator in a Hilbert space ,

many of the results of this section become simpler and can be

**proved**moredirectly by other methods . With these hypotheses considerable extension is

possible .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero