## Linear Operators: General theory |

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Page 20

This notion is written symbolically a„ -> a, or lim^,, an — a. A sequence {a„} is said

to be convergent if an -> a for some a. A sequence {a„} in a metric space is a

...

This notion is written symbolically a„ -> a, or lim^,, an — a. A sequence {a„} is said

to be convergent if an -> a for some a. A sequence {a„} in a metric space is a

**Cauchy sequence**if limm „ n(a„,, a„) = 0. If even'**Cauchy sequence**is convergent,...

Page 68

Every sequence {xn} such that {x*xn} is a

e X* is called a weak

complete if every weak

topology ...

Every sequence {xn} such that {x*xn} is a

**Cauchy sequence**of scalars for each x*e X* is called a weak

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

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

Page 122

We will first show that {gn} is a

iii), a set Ee with v(/u, Ee) < oo such that |6,-ft.|i ^ 2e +jE |g„(«)-gm(*)|»G"> ds), n,

m ^ 1. Thus to see that the sequence {g„} is a

We will first show that {gn} is a

**Cauchy sequence**in LX(S). For s > 0 there is, by (iii), a set Ee with v(/u, Ee) < oo such that |6,-ft.|i ^ 2e +jE |g„(«)-gm(*)|»G"> ds), n,

m ^ 1. Thus to see that the sequence {g„} is a

**Cauchy sequence**in L^S) it will ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero