## Linear Operators: General theory |

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Page 34

The

satisfy the following conditions: (i) a(bc) = (ab)c, a,b,c e G; (ii) there is an clement

e in G, called the identity or the unit of G, such that ae = ea = a for every a in G, (iii

) ...

The

**element**ab is called the product of a and b. The product ab is required tosatisfy the following conditions: (i) a(bc) = (ab)c, a,b,c e G; (ii) there is an clement

e in G, called the identity or the unit of G, such that ae = ea = a for every a in G, (iii

) ...

Page 40

If R is a ring with unit <*, then an

case R contains a (right, left) inverse y for x, i.e., we have (xy = e, yx = e) xy = yx =

e. If x is regular, its unique inverse is denoted by x~x. An

If R is a ring with unit <*, then an

**element**x in R is called {right, left) regular in R incase R contains a (right, left) inverse y for x, i.e., we have (xy = e, yx = e) xy = yx =

e. If x is regular, its unique inverse is denoted by x~x. An

**element**which is not ...Page 335

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

ordered under the partial ordering of L is at ... between

mean that a Q b and that each

bound for a.

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

**elements**of L which is well-ordered under the partial ordering of L is at ... between

**elements**a, b in W tomean that a Q b and that each

**element**x which is in b but not a is an upperbound for a.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero