## Linear Operators: General theory |

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Page 161

u e ba. For £ > 0 choose nt so that \/u„—fim | ^ e for to, n 3: ne. Then fi(E)—/x„(E) =

lim^^ (fim(E)—fi„(E)), from which it follows that I/*— /*„j ^ e for n 3: n,. Hence ju„ ft,

which proves that

u e ba. For £ > 0 choose nt so that \/u„—fim | ^ e for to, n 3: ne. Then fi(E)—/x„(E) =

lim^^ (fim(E)—fi„(E)), from which it follows that I/*— /*„j ^ e for n 3: n,. Hence ju„ ft,

which proves that

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

for any E e E and all n = 1, 2 The right hand side is independent of n,

contradicting the supposition that there was a set Gn e E with (//„((?„) | > n for

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

Theorem.

for any E e E and all n = 1, 2 The right hand side is independent of n,

contradicting the supposition that there was a set Gn e E with (//„((?„) | > n for

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

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

Page 340

18 Show that the space bs is isometrically isomorphic with the space l^. Show

how this isomorphism ... 17 Show that a sequence {k„} of elements of

converge weakly to an element X e

negative ...

18 Show that the space bs is isometrically isomorphic with the space l^. Show

how this isomorphism ... 17 Show that a sequence {k„} of elements of

**ba**(**S**, Z)converge weakly to an element X e

**ba**(**S**, Z) if and only if there exists a non-negative ...

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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