## Linear Operators: General theory |

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Page 484

Linear combinations of

product of a

, U be ...

Linear combinations of

**weakly compact**operators are**weakly compact**. Theproduct of a

**weakly compact**linear operator and a bounded linear operator is**weakly compact**. Proof. Let T, V e B(X, <$) ), WeB(ty.Q), VcB&X), x,pe4>, and let T, U be ...

Page 494

From IV. 10.2 we conclude that T* maps the unit sphere of X* into a conditionally

Theorem 4.8 this implies that T is a

From IV. 10.2 we conclude that T* maps the unit sphere of X* into a conditionally

**weakly compact**set of rca(S), and therefore T* is a**weakly compact**operator. ByTheorem 4.8 this implies that T is a

**weakly compact**operator. Q.E.D. 4 Theorem.Page 507

Let (S, 27, u) be a a-finite positive measure space, and let T be a

operator on L^S, 27, p.) to a separable subset of the B-space X. Then there exists

a fi- essentially unique bounded measurable function #(•) on S to a weakly ...

Let (S, 27, u) be a a-finite positive measure space, and let T be a

**weakly compact**operator on L^S, 27, p.) to a separable subset of the B-space X. Then there exists

a fi- essentially unique bounded measurable function #(•) on S to a weakly ...

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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 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