## Linear Operators, Part 1 |

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Page 481

Let E , and E , be commuting

, i = 1 , 2 , then ( a ) E , E , = E , if and only if M , CMı , or equivalently , if and only if

N C Nz ; ( b ) if E = E + E , -E , E2 , then E is a

Let E , and E , be commuting

**projections**in a B - space X. = EzX , N ; = ( I – E ; ) X, i = 1 , 2 , then ( a ) E , E , = E , if and only if M , CMı , or equivalently , if and only if

N C Nz ; ( b ) if E = E + E , -E , E2 , then E is a

**projection**with EX = sp { M , UM ...Page 482

Thus , the family of all

... We conclude this section with a few remarks on

Let E be a

Thus , the family of all

**projections**in B ( X ) form a partially ordered set . Any two... We conclude this section with a few remarks on

**projections**in Hilbert space .Let E be a

**projection**in Hilbert space H , and let E * be its Hilbert space adjoint .Page 514

19 If E is a

dimensional range . 20 A

compact . 21 A linear mapping E such that E2 = E is a

bounded ) ...

19 If E is a

**projection**with n dimensional range , then E * is a**projection**with ndimensional range . 20 A

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

**projection**( i.e. , isbounded ) ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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