## Linear Operators: General theory |

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

in a metric space is a

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

in a metric space is a

**Cauchy sequence**if limm „ g{am, a„) = 0. If every**Cauchy****sequence**is convergent, a metric space is said to be complete. The next three ...Page 68

which converges weakly to a point in X. Every sequence {x„} such that {x*xn} is a

The space X is said to be weakly complete if even- weak

which converges weakly to a point in X. Every sequence {x„} such that {x*xn} is a

**Cauchy sequence**of scalars for each x* e X* is called a rcecJ:**Cauchy sequence**.The space X is said to be weakly complete if even- weak

**Cauchy sequence**...Page 122

p- We will first show that {gn} is a

(iii), a set Ee with v(/u, Ee) < oo such that |gB— gmli S2e + I \gn(s) - gm{s)\v(n, ds),

n,m ^ 1. Thus to sec that the sequence {g„} is a

p- We will first show that {gn} is a

**Cauchy sequence**in L1(5). For e > 0 there is, by(iii), a set Ee with v(/u, Ee) < oo such that |gB— gmli S2e + I \gn(s) - gm{s)\v(n, ds),

n,m ^ 1. Thus to sec that the sequence {g„} is a

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

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact