## Linear Operators, Part 1 |

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Page 145

A sequence of functions (f,\ defined on S with values in 3 converges u-

for each e > 0 there is a set E e X such that v(u, E) < e and such that {f,}

converges

function f if ...

A sequence of functions (f,\ defined on S with values in 3 converges u-

**uniformly**iffor each e > 0 there is a set E e X such that v(u, E) < e and such that {f,}

converges

**uniformly**on S– E. The sequence {fi} converges u-**uniformly**to thefunction f if ...

Page 314

Q.E.D. 12 THEOREM. A subset K of bass, 2) is weakly sequentially compact if

and only if there exists a non-negative u in bass, 2) such that lim Å(E) = 0 a(E)—-

0

...

Q.E.D. 12 THEOREM. A subset K of bass, 2) is weakly sequentially compact if

and only if there exists a non-negative u in bass, 2) such that lim Å(E) = 0 a(E)—-

0

**uniformly**for Že K. PRoof. Let K C bass, 2) be weakly sequentially compact and...

Page 360

which vanishes in a neighborhood of a point p the sequence (S,f)(a) converges to

zero

then the convergence of Sof for a given c.o.m. system is localized if and only if ...

which vanishes in a neighborhood of a point p the sequence (S,f)(a) converges to

zero

**uniformly**for a in some neighborhood of p. 21 Show that if |s; E,(c, z)dz| < M,then the convergence of Sof for a given c.o.m. system is localized if and only if ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero