## Linear Operators: General theory |

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Page 289

there is a functional y* e L* such that ***(*♢) = y*(g) when g and x* are connected,

as in Theorem 1, by the formula «*/=» Js/(*te(«)Ai(*). Applying Theorem 1 once ...

**Proof**. Let x** e (£*)*. By Theorem 1, L* is isometrically isomorphic to L„, so thatthere is a functional y* e L* such that ***(*♢) = y*(g) when g and x* are connected,

as in Theorem 1, by the formula «*/=» Js/(*te(«)Ai(*). Applying Theorem 1 once ...

Page 415

Hence k0 e p — A, and thus p e A -\-k0 Q A -\-K. Q.E.D. Since the commutativity of

the group G is not essential to the

topological groups. 4 Lemma. For arbitrary sets A, B in a linear space i: (i) co(a^l)

...

Hence k0 e p — A, and thus p e A -\-k0 Q A -\-K. Q.E.D. Since the commutativity of

the group G is not essential to the

**proof**, the same result holds for non-Abeliantopological groups. 4 Lemma. For arbitrary sets A, B in a linear space i: (i) co(a^l)

...

Page 699

2 Therefore 2VtJo □e-V* poo ^ \» V* Jo 2y2 which proves (*) and completes the

CPk. For technical reasons occurring later the following lemma is stated for ...

2 Therefore 2VtJo □e-V* poo ^ \» V* Jo 2y2 which proves (*) and completes the

**proof**of the lemma. Q.E.D. We shall now state and prove the lemma referred to asCPk. For technical reasons occurring later the following lemma is stated for ...

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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