## Linear Operators: General theory |

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Page 6

Thus, in both cases /(z) 5* which proves that v e P, and that P has the property (iii)

. Thus P is an admissible subset of A, and so P = A. Therefore, by (v) it is seen

that for any two

is ...

Thus, in both cases /(z) 5* which proves that v e P, and that P has the property (iii)

. Thus P is an admissible subset of A, and so P = A. Therefore, by (v) it is seen

that for any two

**elements**x, z in A, either 2 g x, or z ^ f(x) 2; x, which shows that Ais ...

Page 34

The binary operation fi is often written as [i(a, b) = ab, and, when this notation is

used, it is called multiplication. The

The product ab is required to satisfy the following conditions: (i) a(bc) = (ab)c. a, b

, ...

The binary operation fi is often written as [i(a, b) = ab, and, when this notation is

used, it is called multiplication. The

**element**ab is called the product of a and b.The product ab is required to satisfy the following conditions: (i) a(bc) = (ab)c. a, b

, ...

Page 335

Let L be a a-compkte lattice in which every set of

ordered under the partial ordering of L is at most countable. Then L is complete

and every subset A of L lias a least upper bound which is the least upper bound

of a ...

Let L be a a-compkte lattice in which every set of

**elements**of L which is well-ordered under the partial ordering of L is at most countable. Then L is complete

and every subset A of L lias a least upper bound which is the least upper bound

of a ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero