## Linear Operators: General theory |

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Page 412

The

The

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

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 .Page 421

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

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

**linear functionals**on X which are continuous in the I topology are precisely the functionals in l . The proof of Theorem 9 will be based on the following lemma .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero