## Linear Operators: General theory |

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Page 483

If T is

is compact and hence closed in the ?)* topology of?)**. Thus if T is

If T is

**weakly compact**, then TS is compact in the 9)* topology of ^) and thus x(TS)is compact and hence closed in the ?)* topology of?)**. Thus if T is

**weakly****compact**, (i) yields According to Theorem V.4.5, S1 = S**, and so T**S** Q x{TS).Page 494

From IV. 10.2 we conclude that T* maps the unit sphere of 36* 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 36* 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, E, p) be a a- finite positive measure space, and let T be a

operator on Lt(S, E, p) to a separable subset of the B-space £. Then there exists a

p-esscntially unique bounded measurable function x(-) on S to a weakly ...

Let (S, E, p) be a a- finite positive measure space, and let T be a

**weakly compact**operator on Lt(S, E, p) to a separable subset of the B-space £. Then there exists a

p-esscntially unique bounded measurable function x(-) 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 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