Linear Operators: General theory |
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Page 21
... 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 . If a topological space ...
... 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 . If a topological space ...
Page 450
... dense subset of its boundary . = -T ( x , y ) PROOF . Let K be the convex set . It will be shown that T ( x , −y ) , y e X , for x in a dense subset Z of X. The set K contains some sphere S ( 0 , 1 / N ) about the origin . This clearly ...
... dense subset of its boundary . = -T ( x , y ) PROOF . Let K be the convex set . It will be shown that T ( x , −y ) , y e X , for x in a dense subset Z of X. The set K contains some sphere S ( 0 , 1 / N ) about the origin . This clearly ...
Page 451
... dense in X. Suppose that Z is not dense in X , and that pZ . Then some sphere S ( p , ε ) does not intersect Z. If SS ( p , ε / 2 ) , then SZ = 4 . Hence U U SZ , S. It follows from Theorem I.6.9 that some set Z , contains an open set ...
... dense in X. Suppose that Z is not dense in X , and that pZ . Then some sphere S ( p , ε ) does not intersect Z. If SS ( p , ε / 2 ) , then SZ = 4 . Hence U U SZ , S. It follows from Theorem I.6.9 that some set Z , contains an open set ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ