## Linear Operators, Part 1 |

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Page 20

A sequence {an} is said to be convergent if an -- a for some a. A sequence {an} in

a metric space is a

A sequence {an} is said to be convergent if an -- a for some a. A sequence {an} in

a metric space is a

**Cauchy sequence**if limo., Q(a, , a,) = 0. If every**Cauchy****sequence**is convergent, a metric space is said to be complete. The next three ...Page 68

Every sequence {a,} such that {r+a,} is a

e 3." is called a weak

complete if every weak

topology is ...

Every sequence {a,} such that {r+a,} is a

**Cauchy sequence**of scalars for each a'e 3." is called a weak

**Cauchy sequence**. The space 3 is said to be weaklycomplete if every weak

**Cauchy sequence**has a weak limit. In Chapter V, atopology is ...

Page 122

Let 1 < p < 00; let {f,} be a sequence of functions in Lp(S, 2, u, 3) and let f be a

function on S to 3. Then f is ... Thus to see that the sequence {gn} is a

Hence ...

Let 1 < p < 00; let {f,} be a sequence of functions in Lp(S, 2, u, 3) and let f be a

function on S to 3. Then f is ... Thus to see that the sequence {gn} is a

**Cauchy****sequence**in Li(S) it will suffice to prove that it is a**Cauchy sequence**in Li (E.).Hence ...

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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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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