## Linear Operators: General theory |

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Page 136

(Hahn

set function ft on a field E lias a countably additive non-negative

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

(Hahn

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

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

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

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

a-field E* is known as the Lebesgue

the measure space (S, E*, fi) is the Lebesgue

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

**extension**of ft. Thea-field E* is known as the Lebesgue

**extension**(relative to /x) of the a-field E, andthe measure space (S, E*, fi) is the Lebesgue

**extension**of the measure space ...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 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