## Linear Operators: General theory |

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Hence , by Theorem 7 , there is a maximal well - ordered

= E for if x is in E but not in E , the ... 2 If a family E of

property that A e E if and only if every finite

Hence , by Theorem 7 , there is a maximal well - ordered

**subset**E . of E . Now E ,= E for if x is in E but not in E , the ... 2 If a family E of

**subsets**of a set has theproperty that A e E if and only if every finite

**subset**of A belongs to E , then E ...Page 439

Let K be a

A C K is said to be an extremal

1 - a ) k , , 0 ) < a < 1 , of two points of K is in A only if both ky and kq are in A ...

Let K be a

**subset**of a real or complex linear vector space X . A non - void**subset**A C K is said to be an extremal

**subset**of K if a proper convex combination ak , + (1 - a ) k , , 0 ) < a < 1 , of two points of K is in A only if both ky and kq are in A ...

Page 440

totally ordered subfamily of A , the non - void set 0 A , is a closed extremal

of K which furnishes a lower bound for A . It follows by Zorn ' s lemma that A

contains a minimal element A . . Suppose that A , contains two distinct points p

and q ...

totally ordered subfamily of A , the non - void set 0 A , is a closed extremal

**subset**of K which furnishes a lower bound for A . It follows by Zorn ' s lemma that A

contains a minimal element A . . Suppose that A , contains two distinct points p

and q ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

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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