## Linear Operators: General theory |

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Page 161

u e ba. For e > 0 choose ne so that \fin— fim\ e for to, n 2: n,. Then /t(E)—/u„(E) —

limBl_V00 (/*„,(£) — from which it follows that |yU— ^ e for n wt. Hence /in -»□ fx,

which proves that

u e ba. For e > 0 choose ne so that \fin— fim\ e for to, n 2: n,. Then /t(E)—/u„(E) —

limBl_V00 (/*„,(£) — from which it follows that |yU— ^ e for n wt. Hence /in -»□ fx,

which proves that

**ba**(**S**, E, 36) is complete. It follows, therefore, that**ba**(**S**, E, 36) ...Page 311

The space

also weakly complete. Proof. Consider the closed subspace B(S, Z) of B(S).

According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S1

such that ...

The space

**ba**(**S**,Z) is weakly complete. If S is a topological space, the rba(S) isalso weakly complete. Proof. Consider the closed subspace B(S, Z) of B(S).

According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S1

such that ...

Page 340

16 Let S be a completely regular topological space. Show that C(S) is separable

if and only if S is compact and metric. 17 Show that a sequence {A„} of elements

of

16 Let S be a completely regular topological space. Show that C(S) is separable

if and only if S is compact and metric. 17 Show that a sequence {A„} of elements

of

**ba**(**S**, E) converge weakly to an element A e**ba**(**S**, E) if and only if there exists ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero