## Linear Operators: General theory |

### From inside the book

Results 1-3 of 14

Page 376

13 13 . 14 6 . 29 6 . 29 6 . 14 Weak Cauchy Sequences 13 . 13 13 . 14 13 . 43 13 .

43 13 . 40 Weak Convergence to x 6 . 4 , 6 . 12 , 6 . 31 13 . 13 13 . 14 6 . 31 6 . 31

Y Convergence in X = Y *

13 13 . 14 6 . 29 6 . 29 6 . 14 Weak Cauchy Sequences 13 . 13 13 . 14 13 . 43 13 .

43 13 . 40 Weak Convergence to x 6 . 4 , 6 . 12 , 6 . 31 13 . 13 13 . 14 6 . 31 6 . 31

Y Convergence in X = Y *

**meaning**- less**meaning**- meaningless less 13 .Page 379

7

meaningless 13 . 37 13 . 38 13 . 37 13 . 39 13 . 45 11 . 1 13 . 47 Strongly

Compact Sets 13 . 37 6 . 29 Weakly Compact Sets 4 . 7

meaningless 13 . 37 13 .

7

**meaning**- less meaningless Reflexivity no no yes 4 . 6 meaninglessmeaningless 13 . 37 13 . 38 13 . 37 13 . 39 13 . 45 11 . 1 13 . 47 Strongly

Compact Sets 13 . 37 6 . 29 Weakly Compact Sets 4 . 7

**meaning**- lessmeaningless 13 . 37 13 .

Page 534

L ( S , & , u ) has a known

how to define L . ( S , E , u ) . For 1 Eps 00 , L , is a B - space , but for 0 < p < 1 , L ,

is merely an F - space ( cf . Exercise III . 9 . 30 , where , however , the symbol to ...

L ( S , & , u ) has a known

**meaning**; in the course of this section we will also seehow to define L . ( S , E , u ) . For 1 Eps 00 , L , is a B - space , but for 0 < p < 1 , L ,

is merely an F - space ( cf . Exercise III . 9 . 30 , where , however , the symbol to ...

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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