## Linear Operators: General theory |

### From inside the book

Results 1-3 of 85

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 V 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 421

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

functionals in T . 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 & which are continuous in the I topology are precisely thefunctionals in T . The proof of Theorem 9 will be based on the following lemma .

10 LEMMA .

Page 452

If A is a subset of X , and p is in A , then there exists a non - zero continuous

by A is not dense in X . PROOF . If q¢ K , then , by 2 . 12 we can find a functional ...

If A is a subset of X , and p is in A , then there exists a non - zero continuous

**linear functional**tangent to A at p if and only if the cone B with vertex p generatedby A is not dense in X . PROOF . If q¢ K , then , by 2 . 12 we can find a functional ...

### What people are saying - Write a review

User Review - Flag as inappropriate

i want to read

### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

5 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 Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero