## Linear Operators: General theory |

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Page 136

(Hahn

set function pi on a field E has a countably additive non-negative

a-field determined by E. If p. is a-finite on E then this

(Hahn

**extension**) Every countably additive non-negative extended real valuedset function pi on a field E has a countably additive non-negative

**extension**to thea-field determined by E. If p. is a-finite on E then this

**extension**is unique. Proof.Page 143

Then the function fx with domain E* is known as the Lebesgue

The cr-field E* is known as the Lebesgue

, and the measure space (S, E*, /n) is the Lebesgue

Then the function fx with domain E* is known as the Lebesgue

**extension**of fi.The cr-field E* is known as the Lebesgue

**extension**(relative to yu) of the a-field E, and the measure space (S, E*, /n) is the Lebesgue

**extension**of the measure ...Page 554

showed that this operation is linear and isometric for each closed linear manifold

...

**Extension**of Linear Transformation. Taylor [1] studied conditions under which the**extension**of linear functionals will be a uniquely defined operation. Kakutani [6]showed that this operation is linear and isometric for each closed linear manifold

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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