## Linear Operators: General theory |

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Page 98

is the union of some of the sets Ft. Then Elfi(Ej)\ ^ E\{i(F,)\ and we have v(E) = lim

E\n(Ei)\. ... E)+v((i, F). Thus if

F). If v(/x, E(j F) < oo there are finite sequences {Es}, {Ft} of disjoint sets in E with ...

is the union of some of the sets Ft. Then Elfi(Ej)\ ^ E\{i(F,)\ and we have v(E) = lim

E\n(Ei)\. ... E)+v((i, F). Thus if

**v**(**fi**, E\j F) = oo it follows that v(p, E\jF) = v{/u, E)~ v([i,F). If v(/x, E(j F) < oo there are finite sequences {Es}, {Ft} of disjoint sets in E with ...

Page 116

Then there is a set A e Z with

seS and g» =-= 0 for s 4 A. Thus if v(ft, E) < e/A/, EtZ, v(G, E) = j£ \g(s)\

\g(») ~ ge(«)K«. ds) + jAE \g.{sMf*>ds) <e +Mv{f.i,AE) < 2c, proving (b). Since i-(G

...

Then there is a set A e Z with

**v**(**fi**, A) < oo and a constant M with |g»| < M for allseS and g» =-= 0 for s 4 A. Thus if v(ft, E) < e/A/, EtZ, v(G, E) = j£ \g(s)\

**v**(**fi**, ds) ^ jE\g(») ~ ge(«)K«. ds) + jAE \g.{sMf*>ds) <e +Mv{f.i,AE) < 2c, proving (b). Since i-(G

...

Page 157

The space E(fi) is therefore a complete metric space. If X is an additive vector or

scalar valued function on E which is //-continuous, then X is defined and

continuous on the metric space E(p). To see this note first that the identities

EA ...

The space E(fi) is therefore a complete metric space. If X is an additive vector or

scalar valued function on E which is //-continuous, then X is defined and

continuous on the metric space E(p). To see this note first that the identities

**v**(**fi**,EA ...

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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