## Linear Operators: General theory |

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Page 464

(3) For every f e T, sup^ &f(x) < oo; moreover, each linear functional 0 on r which

If K(, f < 6, 6 a limit ordinal, is a monotone decreasing trans- finite sequence of ...

(3) For every f e T, sup^ &f(x) < oo; moreover, each linear functional 0 on r which

**satisfies**inf 9tf{x) g ... f e r xiK x(K also**satisfies**#(/) = /(*,>). f*r for some x0 e K. (4)If K(, f < 6, 6 a limit ordinal, is a monotone decreasing trans- finite sequence of ...

Page 495

We now prove that 930 is an algebra under the natural product (fg)(s) = f(s)g(s),

seS. To see this, let A be a fixed element of C(S) and let 93(A) = {/ e 930|/A e 930}

. Evidently 93(A) Q 930, and it is clear that 93(A)

We now prove that 930 is an algebra under the natural product (fg)(s) = f(s)g(s),

seS. To see this, let A be a fixed element of C(S) and let 93(A) = {/ e 930|/A e 930}

. Evidently 93(A) Q 930, and it is clear that 93(A)

**satisfies**properties (i) and ...Page 557

Consequently, the product /?j of all the factors {X—kiY* in R such that A, € a(T),

still

where f}{ = min (a,, v(A,)),

zero of ...

Consequently, the product /?j of all the factors {X—kiY* in R such that A, € a(T),

still

**satisfies**= 0. In the same way, the product i?2 of all the factors (A — %if',where f}{ = min (a,, v(A,)),

**satisfies**R2(T) = 0. Since any polynomial P having azero of ...

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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