## Linear Operators: General theory |

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Page 161

p e ba. For e > 0 choose ne so that \fin— fim\ iS e for m, n 2: ne. Then fi{E)—fin(E)

= lim^^, {nm(E)—fin(E)), from which it follows that \fi— pn\ e for n 2i ns. Hence /in -

> fi, which proves that

p e ba. For e > 0 choose ne so that \fin— fim\ iS e for m, n 2: ne. Then fi{E)—fin(E)

= lim^^, {nm(E)—fin(E)), from which it follows that \fi— pn\ e for n 2i ns. Hence /in -

> fi, which proves that

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

Q.E.D. Next we turn to an investigation of the space

space

tvealdy complete. Proof. Consider the closed subspace B(S, E) of B(S). According

to ...

Q.E.D. Next we turn to an investigation of the space

**ba**(**S**, E). 9 Theorem. Thespace

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

to ...

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 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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### Common terms and phrases

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero