## Linear Operators: General theory |

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Page 40

An element which is not (right, left ) regular is called (right, left) singular. If 0 is a

field, then a set A' is said to be an

space over 0 and if «.(xy) = (ax)y = x(txy), x,yeX, xe0. A right (left, two-sided) ideal

...

An element which is not (right, left ) regular is called (right, left) singular. If 0 is a

field, then a set A' is said to be an

**algebra**over 0 if X is a ring as well as a vectorspace over 0 and if «.(xy) = (ax)y = x(txy), x,yeX, xe0. A right (left, two-sided) ideal

...

Page 44

On the other hand, if B is a Boolean ring with unit denoted by 1, then if x y is

defined to mean x = xy, and x' = 1 +x then B is a Boolean

xy, XAy = xy. Thus the concepts of Boolean

are ...

On the other hand, if B is a Boolean ring with unit denoted by 1, then if x y is

defined to mean x = xy, and x' = 1 +x then B is a Boolean

**algebra**and xv y = x+y+xy, XAy = xy. Thus the concepts of Boolean

**algebra**and Boolean ring with unitare ...

Page 272

We continue our analysis of the space C(S) with a discussion of certain important

special properties related to its structure as an

a well known approximation theorem of Weierstrass, which asserts that a ...

We continue our analysis of the space C(S) with a discussion of certain important

special properties related to its structure as an

**algebra**. One of these properties isa well known approximation theorem of Weierstrass, which asserts that a ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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