## Linear Operators: General theory |

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Page 240

The space bs is the linear space of all sequences x — {a„} of

norm n \x\ = sup I 2<x*| is finite. 11. The space cs is the linear space of all

sequences x = {a„} for which the series *s convergent. The norm is n \x\ = sup I

2at|.

The space bs is the linear space of all sequences x — {a„} of

**scalars**for which thenorm n \x\ = sup I 2<x*| is finite. 11. The space cs is the linear space of all

sequences x = {a„} for which the series *s convergent. The norm is n \x\ = sup I

2at|.

Page 256

If, however, each of the spaces Sj, . . ., X„ are Hilbert spaces then it will always be

understood, sometimes without explicit mention, that X is the uniquely

determined Hilbert space with

is the ...

If, however, each of the spaces Sj, . . ., X„ are Hilbert spaces then it will always be

understood, sometimes without explicit mention, that X is the uniquely

determined Hilbert space with

**scalar**product (iv) xn], [yv. . ., = 2 (*<» where (•, -){is the ...

Page 323

A

sequence {/„} of simple functions such that (i) /„(«) converges to f(s) ^-almost

everywhere; (ii) the sequence {J Efn(s)[/(ds)} converges in the norm of £ for each

EeZ.

A

**scalar**valued measurable function / is said to be integrable if there exists asequence {/„} of simple functions such that (i) /„(«) converges to f(s) ^-almost

everywhere; (ii) the sequence {J Efn(s)[/(ds)} converges in the norm of £ for each

EeZ.

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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