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. Suppose that Z is not dense in X , and that p & Z. Then some sphere S ( p , ε ) does not intersect Z. If S = S ( 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 ...
... dense in X. Suppose that Z is not dense in X , and that p & Z. Then some sphere S ( p , ε ) does not intersect Z. If S = S ( 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 ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ