Linear Operators: General theory |
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Page 11
... LEMMA . If ẞ is a family of subsets of X , and if t is the family of all unions of subfamilies of ẞ , then τ is a topology if and only if ( i ) for every pair U , Veẞ and xe UV there is a Weẞ , such that xe WCUV , and ( ii ) X = Uẞ . 8 ...
... LEMMA . If ẞ is a family of subsets of X , and if t is the family of all unions of subfamilies of ẞ , then τ is a topology if and only if ( i ) for every pair U , Veẞ and xe UV there is a Weẞ , such that xe WCUV , and ( ii ) X = Uẞ . 8 ...
Page 410
... lemma is an obvious consequence of Definition 1 . 2 LEMMA . The intersection of an arbitrary family of convex subsets of the linear space X is convex . As examples of convex subsets of X , we note the subspaces of X , and the subsets of ...
... lemma is an obvious consequence of Definition 1 . 2 LEMMA . The intersection of an arbitrary family of convex subsets of the linear space X is convex . As examples of convex subsets of X , we note the subspaces of X , and the subsets of ...
Page 697
... Lemma 11 will be an inductive one based upon its special case k 1 which has been established in Lemma 6. Since the proof is a rather circuitous one , based upon three auxiliary lemmas , it will perhaps be of some help if we make a ...
... Lemma 11 will be an inductive one based upon its special case k 1 which has been established in Lemma 6. Since the proof is a rather circuitous one , based upon three auxiliary lemmas , it will perhaps be of some help if we make a ...
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