## Linear Operators: General theory |

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Page 140

It can also be extended by defining the measure differently on the basic field E.

We shall illustrate this latter remark by describing here the construction of a

Radon measure on an

It can also be extended by defining the measure differently on the basic field E.

We shall illustrate this latter remark by describing here the construction of a

Radon measure on an

**interval**. 15 Definition. An**interval**is a set of points in the ...Page 141

a„i+1 fe„i+1 ^ c+Łn2 = an, and such that 2 -/(«,) | > «5- <=»>i+i The argument may

be repeated by defining s3 = a„ —c and choosing appropriate points in the

a„i+1 fe„i+1 ^ c+Łn2 = an, and such that 2 -/(«,) | > «5- <=»>i+i The argument may

be repeated by defining s3 = a„ —c and choosing appropriate points in the

**interval**(e, c+e3]. By induction, it is clear that for every integer k — 1, 2 there are ...Page 283

Proof. The proof will require the following two lemmas. 3 Lemma. An almost

periodic function is bounded. Proof of Lemma 3. Since an almost periodic

function / is continuous, the function |/(*)| has a maximum K on the

l). Let x be ...

Proof. The proof will require the following two lemmas. 3 Lemma. An almost

periodic function is bounded. Proof of Lemma 3. Since an almost periodic

function / is continuous, the function |/(*)| has a maximum K on the

**interval**0 sS L(l). Let x be ...

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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