## Linear Operators: General theory |

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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 limm ,( 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 {xn} such that {x*xn} is a

e 3E* 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 3E* is called a weak

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

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

Page 122

Let 1 si p < oo; let {/„} be a sequence of functions in LV(S, E, (i, X) and let f be a

function on S to £. Then f is in Lv ... Assuming (i), (ii), and (iii) let gn = \f„(')\p, g=\f(-)

\p- We will first show that {g„} is a

Let 1 si p < oo; let {/„} be a sequence of functions in LV(S, E, (i, X) and let f be a

function on S to £. Then f is in Lv ... Assuming (i), (ii), and (iii) let gn = \f„(')\p, g=\f(-)

\p- We will first show that {g„} is a

**Cauchy sequence**in L^S). For g > 0 there is, ...### 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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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero