## Linear Operators: General theory |

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of B, and B is said to contain A. This is denoted symbolically by AQ B, or B~2 A.

Two

contains every element a of A. Two

of B, and B is said to contain A. This is denoted symbolically by AQ B, or B~2 A.

Two

**sets**are the same if and only if they ... are taken relative to some**set**b whichcontains every element a of A. Two

**sets**are**disjoint**if their intersection is void.Page 97

Let ft be a set function defined on the field E of subsets of a set S. Then for every

E in E the total variation of u on E, denoted by v(fi, E), is defined as v(n, E) = sup 2

IfiiEJl where the supremum is taken over all finite sequences {£,} of

Let ft be a set function defined on the field E of subsets of a set S. Then for every

E in E the total variation of u on E, denoted by v(fi, E), is defined as v(n, E) = sup 2

IfiiEJl where the supremum is taken over all finite sequences {£,} of

**disjoint sets**...Page 203

p{E) = f(E), Eel" is uniquely defined on the field Ev To see that fi is additive on E1

let Ev E2 be

E1 e 2>, E2 e 2>. Thus, if = U ^2> we nave ®i> -^2 e and hence there are disjoint

...

p{E) = f(E), Eel" is uniquely defined on the field Ev To see that fi is additive on E1

let Ev E2 be

**disjoint sets**in Ev By Lemma 18, there are finite sets nv n2 in A withE1 e 2>, E2 e 2>. Thus, if = U ^2> we nave ®i> -^2 e and hence there are disjoint

...

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact