## 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 Z^- = {a:*|a:* 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 arc complete, show that X is also.17 Definition. If X is a normed linear space, and ZCX, the set Z^- = {a:*|a:* e X*, ...

Page 420

On the other hand, if I is a

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

Q X+ is obviously total. The r topology of X is often called the 9) topology of X. It is

...

On the other hand, if I is a

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

**subspace**r = {fy\y e Y}Q X+ is obviously total. The r topology of X is often called the 9) topology of X. It is

...

Page 436

6 Let X be a B-space. and i\ a

the same as the relative X* topology of Xv 7 Let J be a linear space, and r a total

bounded ...

6 Let X be a B-space. and i\ a

**subspace**of X. Show that the X* topology of i'j isthe same as the relative X* topology of Xv 7 Let J be a linear space, and r a total

**subspace**of X*. Show- that a set A Q X is /'-bounded if and only if f(A) is abounded ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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### Common terms and phrases

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