## Linear Operators: General theory |

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Page 161

H e ba. For e > 0 choose nc so that \[in— fim\ £ for m, n 2^ nc. Then u(E)—n„(E) =

lim^^ (nm(E)—fi„{E)), from which it follows that Jjt« — | :£ e for n 5? ?ie. Hence //„ -

> ft, which proves that

H e ba. For e > 0 choose nc so that \[in— fim\ £ for m, n 2^ nc. Then u(E)—n„(E) =

lim^^ (nm(E)—fi„{E)), from which it follows that Jjt« — | :£ e for n 5? ?ie. Hence //„ -

> ft, which proves that

**ba**(**S**, E, X) is complete. It follows, therefore, that**ba**(**S**, 27, ...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 {An} 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 {An} of elements

of

**ba**(**S**, E) converge weakly to an element A €**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 | 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