## Linear Operators: General theory |

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Page 240

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

norm \x\ = sup I Jail n (-1 is finite. 11. The space cs is the linear space of all

sequences x — {oc„} for which the series a« is convergent. The norm is n \x\ =

sup I ...

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

**scalars**for which thenorm \x\ = sup I Jail n (-1 is finite. 11. The space cs is the linear space of all

sequences x — {oc„} for which the series a« is convergent. The norm is n \x\ =

sup I ...

Page 256

If, however, each of the spaces #j. . . ., 3£„ arc Hilbert spaces then it will always

be understood, sometimes without explicit mention, that £ is the uniquely

determined Hilbert space with

t, ...

If, however, each of the spaces #j. . . ., 3£„ arc Hilbert spaces then it will always

be understood, sometimes without explicit mention, that £ is the uniquely

determined Hilbert space with

**scalar**product (iv) ([xv . . ., x„], [yv. . ., yn]) = £ (xt, yt)t, ...

Page 323

A

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

everywhere; (ii) the sequence {J" Efn(s)fi(ds)} converges in the norm of X for each

E e Z.

A

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

everywhere; (ii) the sequence {J" Efn(s)fi(ds)} converges in the norm of X for each

E e Z.

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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