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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 ; since Zn , i Zn , this implies that some set Z , is not dense . We will show that this is impossible . Suppose that a1e X , x1 Zn , then t ( x , y ) −T ( x , −yn ) for x in some relative neighborhood of x1 . Thus , the ...
... dense in X ; since Zn , i Zn , this implies that some set Z , is not dense . We will show that this is impossible . Suppose that a1e X , x1 Zn , then t ( x , y ) −T ( x , −yn ) for x in some relative neighborhood of x1 . Thus , the ...
Page 842
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40-41 ( 438-439 ) Dense set , definition , I.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 , I.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 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ