## Linear Operators: General theory |

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Page 72

16 Let X be a normed linear space which is not assumed to be complete, and let

3 be a closed

17 Definition. If X is a normed linear space, and ZCX, the set Z1- — {x*\x* e X*, ...

16 Let X be a normed linear space which is not assumed to be complete, and let

3 be a closed

**subspace**of X. Then, if 3 and X/3 are complete, show that X is also.17 Definition. If X is a normed linear space, and ZCX, the set Z1- — {x*\x* e X*, ...

Page 420

On the other hand, if 3c is a

determines the linear functional /„ on X defined by fy(x) = x(y), xel, and the

the "?) topology ...

On the other hand, if 3c is a

**subspace**of ?)+, then each element y e ?)determines the linear functional /„ on X defined by fy(x) = x(y), xel, and the

**subspace**r — {fv\y e F} £ 3£+ is obviously total. The r topology of X is often calledthe "?) topology ...

Page 436

6 Let X be a B-space, and Xx a

the same as the relative X* topology of Xr 7 Let X be a linear space, and 71 a

total

6 Let X be a B-space, and Xx a

**subspace**of X. Show that the Xf topology of Xx isthe same as the relative X* topology of Xr 7 Let X be a linear space, and 71 a

total

**subspace**of X+. Show that a set A Q X is .T-bounded if and only if f(A) is a ...### 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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### Common terms and phrases

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