## Linear Operators: General theory |

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Page 169

5 Show that (i), (ii), and (iii) of Theorem 3.6 imply that / is in L„(S, E, /<) and that |/„

— f\v converges to

bounded. Suppose that the field E is separable under the metric q(E, F) = v(/x,

E A F).

5 Show that (i), (ii), and (iii) of Theorem 3.6 imply that / is in L„(S, E, /<) and that |/„

— f\v converges to

**zero**even if {/„} is a generalized sequence. 6 Let fi bebounded. Suppose that the field E is separable under the metric q(E, F) = v(/x,

E A F).

Page 204

Since does not converge to

dominated convergence theorem (6.16) that there is a point «2 in <SL for which /„

(s? ) is defined for all n and for which the se- quence )} does not converge to

.

Since does not converge to

**zero**and since 0 fn(so^) ^ 1 it follows from thedominated convergence theorem (6.16) that there is a point «2 in <SL for which /„

(s? ) is defined for all n and for which the se- quence )} does not converge to

**zero**.

Page 231

Then for each e > 0, the function z~ef(z) is analytic and uniformly bounded in the

strip, tends to

Hence |z~*/(z)| assumes its maximum value somewhere in the strip. By the ...

Then for each e > 0, the function z~ef(z) is analytic and uniformly bounded in the

strip, tends to

**zero**as y -»• i oo, and is bounded by M on the edges of the strip.Hence |z~*/(z)| assumes its maximum value somewhere in the strip. By the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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