Linear Operators: General theory |
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Page 412
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 ...
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 linear functional on X separates K , and Kz . 12 COROLLARY .
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 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 .
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 .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Common terms and phrases
Akad algebra Amer analytic applied arbitrary assume 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 domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence 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
Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 1 of Linear Operators: General Theory. Spectral Theory, Self Adjoint Operators in Hilbert Space. Spectral Operators. I. II. III, William George Bade Volume 7 of Pure & Applied Mathematics : a Wiley-interscience series of texts, monographs & tracts Volume 7 of Pure and applied mathematics : a series of texts and monographs Volume 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics, ISSN 0079-8185 Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle |
| Editors | William G. Bade, Robert G. Bartle |
| Contributor | Karreman Mathematics Research Collection |
| Edition | reprint |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| ISBN | 0470226056, 9780470226056 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |