## Linear Operators: General theory |

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Page 251

It is clear that E is a

D is an orthogonal

M ...

It is clear that E is a

**projection**, i . e . , E2 = E , and that E is an orthogonal**projection**. It is the uniquely determined orthogonal**projection**with EH = M . For ifD is an orthogonal

**projection**with DH = M then ED = D and , since ( I - D ) H CHOM ...

Page 514

20 A

linear mapping E such that E2 = E is a

if the ranges of E and I - E are closed . 22 Let E be a

...

20 A

**projection**has finite dimensional range if and only if it is compact . 21 Alinear mapping E such that E2 = E is a

**projection**( i . e . , is bounded ) if and onlyif the ranges of E and I - E are closed . 22 Let E be a

**projection**in the B - space X...

Page 515

Nelson Dunford, Jacob T. Schwartz. ( i ) If E is a perpendicular

that its matrix representation is ( aij ) = ( { k = Pik Pik ) , where gk = Pikék , k = 1 , . .

. , r is any orthonormal basis of M = E ( X ) . ( ii ) Let M and N have bases { aq , . .

Nelson Dunford, Jacob T. Schwartz. ( i ) If E is a perpendicular

**projection**, showthat its matrix representation is ( aij ) = ( { k = Pik Pik ) , where gk = Pikék , k = 1 , . .

. , r is any orthonormal basis of M = E ( X ) . ( ii ) Let M and N have bases { aq , . .

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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