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 451
... dense in X , and 00 i φ . 101 Zn , is dense that p ¢ Z. Then some sphere S ( p , e ) does not intersect Z. If S = S ( p , ε / 2 ) , then SZ = 6 . Hence U11 SZ , S. It follows from Theorem I.6.9 that some set Zn , contains an open set ...
... dense in X , and 00 i φ . 101 Zn , is dense that p ¢ Z. Then some sphere S ( p , e ) does not intersect Z. If S = S ( p , ε / 2 ) , then SZ = 6 . Hence U11 SZ , S. It follows from Theorem I.6.9 that some set Zn , contains an open set ...
Page 842
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40 41 ( 438-439 ) Dense set , definition , 1.6.11 ( 21 ) density of simple functions in L 1 ≤ p < ∞ , III.3.8 ( 125 ) density of continuous functions in TM and L ,, III ...
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40 41 ( 438-439 ) Dense set , definition , 1.6.11 ( 21 ) density of simple functions in L 1 ≤ p < ∞ , III.3.8 ( 125 ) density of continuous functions in TM and L ,, III ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ