## 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 K1 , 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 K1 , 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 Kz . 12 COROLLARY .Page 421

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

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 the functionals in T. 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 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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