Linear Operators: General theory |
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Page 21
A set is said to be dense in a topological space X, if its closure is X. It is said to be
nowhere dense if its closure does not contain any open set. A space is separable
, if it contains a denumer- able dense set. 12 Theorem. // a topological space ...
A set is said to be dense in a topological space X, if its closure is X. It is said to be
nowhere dense if its closure does not contain any open set. A space is separable
, if it contains a denumer- able dense set. 12 Theorem. // a topological space ...
Page 451
We wish to prove that Z = n*_j Z„ = H"=1 Ojlj Zn j is dense in X. Suppose that Z is
not dense in X, and that p 4 Z. Then some sphere 5(p, e) does not intersect Z. If S
= S{p, e/2), then SZ = 9S. Hence U"=1 U,*! SZ^ = S. It follows from Theorem 1.6.9
...
We wish to prove that Z = n*_j Z„ = H"=1 Ojlj Zn j is dense in X. Suppose that Z is
not dense in X, and that p 4 Z. Then some sphere 5(p, e) does not intersect Z. If S
= S{p, e/2), then SZ = 9S. Hence U"=1 U,*! SZ^ = S. It follows from Theorem 1.6.9
...
Page 842
1.7 (98) for measures, III.4.7 (128), III. 4.11 (130) Lebesgue decomposition, III. 4.
14 (132) Saks decomposition, IV.9.7 (308) Yosida-Hewitt decomposition, (233 )
De Morgan, rules of, (2) Dense convex sets, V.7.27 (437 ) Dense linear manifolds
, ...
1.7 (98) for measures, III.4.7 (128), III. 4.11 (130) Lebesgue decomposition, III. 4.
14 (132) Saks decomposition, IV.9.7 (308) Yosida-Hewitt decomposition, (233 )
De Morgan, rules of, (2) Dense convex sets, V.7.27 (437 ) Dense linear manifolds
, ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries 84 | 34 |
Copyright | |
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Common terms and phrases
Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero
Bibliographic information
| Title | Linear Operators: General theory Volume 7 of Pure and applied mathematics Volume 1 of Linear Operators, Nelson Dunford |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| Length | 2592 pages |
| Subjects | › › Algebra, Universal Algebras, Linear Linear operators Mathematical analysis Mathematics / Algebra / Linear |
| Export Citation | BiBTeX EndNote RefMan |