## Linear Operators: General theory |

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Page 38

Algebras If

RA form a ring under the operations defined

right ideal

Algebras If

**R**is a ring, then a subset RXQ**R**is called a subring if the elements inRA form a ring under the operations defined

**in R**. A sub- ring /**of R**is called aright ideal

**of R**if it has the additional properties (a) IxQI, xe**R**, (b) (O)^I^**R**. The ...Page 39

x e

so

{left, or two-sided) ideal, if it is contained in no other ideal of the same type.

x e

**R**such that e — ax — xa. Thus every non-zero element**of R**has an inverse,so

**R**is a field. A right (left, or two-sided) ideal in a ring**R**is called a maximal right{left, or two-sided) ideal, if it is contained in no other ideal of the same type.

Page 192

Then gn(

that each function g„ is p-measurable. Thus, there is no loss in generality if we

prove the lemma only for the case of the totally //-measurable functions /„.

Assume ...

Then gn(

**r**) ->• g(**r**) for each**r**and hence, by Corollary 6.14, it is sufficient to showthat each function g„ is p-measurable. Thus, there is no loss in generality if we

prove the lemma only for the case of the totally //-measurable functions /„.

Assume ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero