## Linear Operators: General theory |

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Any statement concerning the points of S is said to hold fi-almost

if fi is understood, simply almost

except for those points s in a //-null set. The phrase "almost

Any statement concerning the points of S is said to hold fi-almost

**everywhere**, or,if fi is understood, simply almost

**everywhere**. or for almost all s in S, if it is trueexcept for those points s in a //-null set. The phrase "almost

**everywhere**" is ...Page 150

(b) // (S, E, /u) is a finite measure space, an almost

sequence of measurable functions is convergent in /x-measure. 14 Corollary. // (S

,£,jj.) is a measure space, and {/„} is a sequence of measurable vector valued ...

(b) // (S, E, /u) is a finite measure space, an almost

**everywhere**convergentsequence of measurable functions is convergent in /x-measure. 14 Corollary. // (S

,£,jj.) is a measure space, and {/„} is a sequence of measurable vector valued ...

Page 676

1 1.2, the sequence A(T, n)h converges almost

Since L„ is dense in Lt we may apply Lemma 5 and Theorem IV. 11.2 again to

see that the sequence A(T, n)f converges almost

Q.E.D. It is ...

1 1.2, the sequence A(T, n)h converges almost

**everywhere**for every h in LP.Since L„ is dense in Lt we may apply Lemma 5 and Theorem IV. 11.2 again to

see that the sequence A(T, n)f converges almost

**everywhere**for every / in LrQ.E.D. It is ...

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact