## Linear Operators: General theory |

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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 418

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

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

on X separates K , and Ką . 12 COROLLARY . If K is a closed convex subset of ...

If Kį , 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 Ką . 12 COROLLARY . If K is a closed convex subset of ...

Page 421

Let X be a linear space , and let l be a total subspace of X + . Then the

functionals in T ' . The proof of Theorem 9 will be based on the following lemma .

Let X be a linear space , and let l be a total subspace of X + . Then the

**linear****functionals**on X which are continuous in the r ' topology are precisely thefunctionals in T ' . The proof of Theorem 9 will be based on the following lemma .

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero