## Linear Operators, Part 1 |

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Page 481

Some useful elementary properties of

following lemmas . ... E , E , , then E is a

– E ) X = Nin Nz ; ( c ) if E = E , E , , then E is a

– E ) X ...

Some useful elementary properties of

**projections**are summarized in thefollowing lemmas . ... E , E , , then E is a

**projection**with EX = sp { M , UM ; } and ( I– E ) X = Nin Nz ; ( c ) if E = E , E , , then E is a

**projection**with EX = M NM , and ( I– E ) X ...

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

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...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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