## 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 LV(S, E, /t) and that \f„

— f\v converges to

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

F).

5 Show that (i), (ii), and (iii) of Theorem 3.6 imply that / is in LV(S, E, /t) and that \f„

— f\v converges to

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

F).

Page 204

Since ^(fj = J"Sa A^K/d^) does not converge to

follows from the dominated convergence theorem (6.16) that there is a point s° in

Sa for which ) is defined for all n and for which the se- '° ) «i' quence {fn(s° )} does

...

Since ^(fj = J"Sa A^K/d^) does not converge to

**zero**and since 0 sS f„(saJ Si 1 itfollows from the dominated convergence theorem (6.16) that there is a point s° in

Sa for which ) is defined for all n and for which the se- '° ) «i' quence {fn(s° )} does

...

Page 452

If A is a subset of X, and p is in A, then there exists a non-

functional tangent to A at p if and only if the cone B with vertex p generated by A

is not dense in X. Proof. If qiK, then, by 2.12 we can find a functional / and a real ...

If A is a subset of X, and p is in A, then there exists a non-

**zero**continuous linearfunctional tangent to A at p if and only if the cone B with vertex p generated by A

is not dense in X. Proof. If qiK, then, by 2.12 we can find a functional / and a real ...

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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