## Linear Operators: General theory |

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Page 72

11) is a

Cauchy sequence in X/3> define a subsequence for which \xk—xk+1-\-Q\ < 2~k, k

= 1, 2 and show that a Cauchy sequence in X can be chosen which maps onto ...

11) is a

**B**-**space**(or an F-spaee) with the metric \x+Q\ = inf \x+z\. "8 (Hint: Given aCauchy sequence in X/3> define a subsequence for which \xk—xk+1-\-Q\ < 2~k, k

= 1, 2 and show that a Cauchy sequence in X can be chosen which maps onto ...

Page 436

Exercises 1 Let X be a linear vector space. Show that X+ is a total space of linear

functionals on X. 2 If X is an infinite dimensional

a linear topological space. Then X has a non-zero continuous linear functional ...

Exercises 1 Let X be a linear vector space. Show that X+ is a total space of linear

functionals on X. 2 If X is an infinite dimensional

**B**-**space**then X+ ^ X*. 8 Let X bea linear topological space. Then X has a non-zero continuous linear functional ...

Page 843

Density of the natural embedding of a

-6 (424--125) Derivative, chain rulefor,III.13.1 (222) existence of, III. 12. 6 (214) of

functions, III.12.7-8 (216 -217), III. 13.3 (222), III.13.6 (223) properties, IV.

Density of the natural embedding of a

**B**-**space**X into X** in the X* topology, V.4.5-6 (424--125) Derivative, chain rulefor,III.13.1 (222) existence of, III. 12. 6 (214) of

functions, III.12.7-8 (216 -217), III. 13.3 (222), III.13.6 (223) properties, IV.

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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