## Linear Operators, Part 1 |

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Page 412

10 LEMMA . The

if it 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 ...

10 LEMMA . The

**linear functional**f separates the subsets M and N of X if and onlyif it 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 ...

Page 421

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

functionals in I . 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 I . 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 f

and ...

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 f

and ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero