## Linear Operators: General theory |

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Page 341

Show that a set K C ba(S, E) is conditionally compact if and only if (i) K is

bounded. ... 22 Let S be a normal topological space and rca(S) the regular

countably additive set functions on the field of

is weakly ...

Show that a set K C ba(S, E) is conditionally compact if and only if (i) K is

bounded. ... 22 Let S be a normal topological space and rca(S) the regular

countably additive set functions on the field of

**Borel sets**in S. Prove that (i) rca(S)is weakly ...

Page 492

In the following, denotes the field of

the closed sets of S. If fi is a function on 38 with values in a fi-space, then as in

Definition IV. 10.3, the symbol ||^||(£) denotes the semi-variation of fi over E e 88 ...

In the following, denotes the field of

**Borel sets**in S, i.e., the t7-field generated bythe closed sets of S. If fi is a function on 38 with values in a fi-space, then as in

Definition IV. 10.3, the symbol ||^||(£) denotes the semi-variation of fi over E e 88 ...

Page 516

38 (Markov) Let S be a non-void set and <f> a function on S to S. A function fi

defined on the family of subsets of S is said ... that there is a regular countably

additive non-negative measure ft defined for all

that ...

38 (Markov) Let S be a non-void set and <f> a function on S to S. A function fi

defined on the family of subsets of S is said ... that there is a regular countably

additive non-negative measure ft defined for all

**Borel sets**in S with the propertiesthat ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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-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