## Linear Operators: General theory |

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Results 1-3 of 83

Page 77

(1)

|2 (Ara„-^m,n+l)| < 00- M m-0 The transformation preserves sums of series (i.e.

2m=o (2n=o^mnfln) = 2n=oağ) if an£i only if the equation (1') iJ^nn = 1 holds for ...

(1)

**converges**for every w; (2) 2„Umn-Amn+1|**converges**for each to; M (3) SUp£„|2 (Ara„-^m,n+l)| < 00- M m-0 The transformation preserves sums of series (i.e.

2m=o (2n=o^mnfln) = 2n=oağ) if an£i only if the equation (1') iJ^nn = 1 holds for ...

Page 145

A sequence of functions {/„} defined on S with values in i

for each e > 0 there is a set E e Z such that v(fi, E) < e and such that {/„}

uniformly on S — E. The sequence {/„}

A sequence of functions {/„} defined on S with values in i

**converges**u-uniformly iffor each e > 0 there is a set E e Z such that v(fi, E) < e and such that {/„}

**converges**uniformly on S — E. The sequence {/„}

**converges**fi-uniformly to tlie function f if ...Page 281

Let A be a dense subset of a compact Hausdorff space S, and suppose that a

sequence {/„} of continuous functions

continuous limit /„. Then {/„}

and every ...

Let A be a dense subset of a compact Hausdorff space S, and suppose that a

sequence {/„} of continuous functions

**converges**at every point of A to acontinuous limit /„. Then {/„}

**converges**to f0 at every point of S if and only if {/„}and every ...

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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