## Linear Operators: General theory |

### From inside the book

Results 1-3 of 78

Page 412

The

separates the subsets M - N and { 0 } of X. The proof is elementary , and is left to

the reader . In dealing with subspaces , it is often convenient to make use of the

following ...

The

**linear functional**f separates the subsets M and N of X if and only if itseparates the subsets M - N and { 0 } of X. The proof is elementary , and is left to

the reader . In dealing with subspaces , it is often convenient to make use of the

following ...

Page 418

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero continuous

convex subset of ...

If Ky , K , are disjoint closed , convex subsets of a locally convex linear

topological space X , and if K , is compact , then some non - zero continuous

**linear functional**on X separates K , and Kg . 12 COROLLARY . If K is a closedconvex subset of ...

Page 421

Let X be a linear space , and let I ' be a total subspace of x * . Then the

functionals in C. The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

Let X be a linear space , and let I ' be a total subspace of x * . Then the

**linear****functionals**on X which are continuous in the I topology are precisely thefunctionals in C. The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

### Other editions - View all

### Common terms and phrases

Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero