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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 . PROOF . Let K be the convex set . It will be shown that -7 ( x , y ) 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 . PROOF . Let K be the convex set . It will be shown that -7 ( x , y ) 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 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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ